Browse Limits & Continuity problems Studdy Solved!

Problem 601

Find $\lim _{x \rightarrow+\infty} \frac{2 x+5}{x^{2}-7 x+3}$
Add your answer Integer, decimal, or E notation allowed

See Solution

Problem 602

Find $\lim _{x \rightarrow 4} \frac{x^{2}-x-12}{x-4}$
Add your answer Integer, decimal, or E notation allowed

See Solution

Problem 603

Determine where $\mathrm{f}(\mathrm{x})$ is continuous. $f(x)=\sin ^{-1} 2 x$ (A) $-1 / 2$ and $1 / 2$ (B) 1 and - $1 / 2$ (C) -1 and 0 (D) -1 and -3

See Solution

Problem 604

Is the function f such that $f(x)=\frac{1}{x}$ for $\mathrm{x}>0$ and $\mathrm{f}(0)=0$ continuous over $[0,1]$ ? (A) no (B) yes (C) maybe (D) none of the answer

See Solution

Problem 605

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{4 x+49}-7}{x}=$ $\square$

See Solution

Problem 606

Exercice 5 Considérons la fonction $f$ définie par: $f(x)=\frac{x^{2}-2 x+3}{x+3}$
1. Déterminer le domaine de définition $D_{f}$ de $f$ .
2. Calculer les limites aux bornes de $f$ .
3. Interpréter les limites aux bornes réelles.
4. Déterminer les $a, b$ et $c$ tels que $f(x)=a x+b+\frac{c}{x+3}$ .
5. En déduire que la droite $(D)$ d'équation $(D): y=x-5$ est une asymptote oblique pour $\left(C_{f}\right)$ .

See Solution

Problem 607

For the following limits, (a) Determine the Indeterminate form. (b) Compute the limit using L'Hopital's Rule (i) $\lim _{x \rightarrow 0} \frac{\cos (m x)-\cos (n x)}{x^{2}}$ where $\mathrm{n} \& \mathrm{~m}$ are even interger constant.
Hint: Utilitze the Unit Circle for when n or $\mathrm{m}=2,4,6, \ldots$ (ii) $\lim _{x \rightarrow 0^{+}} \sin (x) \ln (x)$ (iii) $\lim _{x \rightarrow 0^{+}} \frac{x}{e^{x}-1}$

See Solution

Problem 608

ابحث في اتصال ق ( س) عند س = 1 ، وغند س = ب من جهة اليسار وايضا ابحث الاتصال على الفترة ( . ، )

See Solution

Problem 609

Теорема 3.2.12. Любой отрезок $[a, b] \subsetneq \mathbb{R}$ конечной длины $a<b$ , является компактным множеством.
Доказательство. Доказывать будем от противного. Долустим, что существует такое покрытие $\left\{\vartheta_{\alpha}\right\}_{\alpha \in A}^{\gamma}$ открытыми множествами из $\mathbb{R}$ для отрезка $I=[a, b]$ , что из него нельзя выбрать конечное подпокрытие.
Итак, пусть $I \subseteq \bigcup_{\alpha \in A} \mathcal{U}_{\alpha}$ и из этого покрытия нельзя выбрать конечное подпокрытие, которое бы покрыло $I$ . Разобьём $I$ пополам т.е представим его так: $[a, b]=\left[a, \frac{a+b}{2}\right] \cup\left[\frac{a+b}{2}, b\right] .$
По условию $I$ нельзя покрыть конечным числом множеств из $\left\{U_{\alpha}\right\}_{\alpha \in A}$ , тогда хотя бы один из полученных отрезков, обозначим его через $I_{1}$ тоже нельзя покрыть конечным числом множеств из покрытия $\left\{\bigcup_{\alpha}\right\}_{\alpha \in A}$ . Иначе бы оба полученных отрезка покрывались бы конечным числом множества из покрытия, а тогда и I покрывался бы конечным числом множеств из этого покрытия, что и показывала бы компактность $I$ .
Разобьём теперь отрезок $I_{1}$ аналогичным образом на два равных отрезка. Так как $I_{1}$ нельзя покрыть конечным числом множеств из покрытия $\left\{थ_{\alpha}\right\}_{\alpha \in A}$ , то найдётся хотя бы один, скажем, $I_{2}$ , из только что полученных, который тоже нельзя покрыть конечным числом множеств. Будем повторять эту процедуру каждый раз, пусть $I_{k}=\left[a_{k}, b_{k}\right], k \geq 1$ . В результате мы получаем бесконечную цепь вложенных друг в друга отрезков $I \supsetneq I_{1} \supsetneq I_{2} \supsetneq \ldots,$
каждый из которых нельзя покрыть конечным числом элементов множества $\left\{U_{\alpha}\right\}_{\alpha \in A}$ . Более того, их длины строго уменьшаются (каждый из отрезков по длине в два раза меньше, чем предыдущий). Тогда по Лемме о вложенных отрезках (Лемма 1.5.6), существует такая точка $c \in I$ , что $c \in \bigcap_{k \geq 1} I_{k}$ , которая есть предел для последовательности их концов; $\lim _{k \rightarrow \infty} a_{k}=c=\lim _{n \rightarrow \infty} b_{n} .$
Тогда для любого $\varepsilon>0$ найдётся такой номер $N$ , что при $k \geq N$ все $a_{k}, b_{k} \in(c-\varepsilon, c+\varepsilon)$ , а по построению это значит, что и все $I_{k} \subseteq(c-\varepsilon, c+\varepsilon), k \geq N$ .
С другой стороны, так как имеется покрытие $\left\{\vartheta_{\alpha}\right\}_{\alpha \in A}$ этого отрезка $I$ , то найдётся хотя бы одно открытое множество $U_{\alpha}$ такое, что $с \in U_{\alpha}$ , а так как оно открытое, то для точки $с$ можно найти $\varepsilon$ -окрестность $(c-\varepsilon, c+\varepsilon) \subseteq U_{\alpha}$ .
Таким образом, мы получаем, что для всех $k \geq N$ есть включения $I_{k} \subseteq(c-\varepsilon, c+\varepsilon) \subseteq U_{\alpha},$
но это значит, что все отрезки $I_{k}$ покрываются всего одним открытым множеством $U_{\alpha}$ , что противоречит построению отрезков $I_{k}$ , т.е. такое построение невозможно, что и означает компактность отрезка $I$ . 58
Теперь у нас всё готово, чтобы описать компактные множества в $\mathbb{R}$ .

See Solution

Problem 610

$\lim [x \rightarrow \infty]\left(\sqrt{x^{2}+3 x+2}+x+1\right)$

See Solution

Problem 611

$\lim _{x \rightarrow 0}\left[x^{-7}\left(\sin x-\frac{x^{5}}{120}+\frac{x^{3}}{6}-x\right)\right]$

See Solution

Problem 612

4.

See Solution

Problem 613

5) $\lim _{x \rightarrow-\infty}-\frac{16 x}{x^{2}+16}$

See Solution

Problem 614

$\lim _{x \rightarrow+\infty} 1+x-x \ln x$

See Solution

Problem 615

Find the secant line slope for $f(x)=\sqrt[3]{x}-11$ at $x_1=750$ and $x_2=6$ .

See Solution

Problem 616

Find the limit: $\lim _{x \rightarrow 0}(\cot (6 x) \cdot \sin (2 x))$ .

See Solution

Problem 617

Find the left-hand and right-hand limits of $f(x) = \sin(x)$ as $x$ approaches 1, and the limit as $x$ approaches -1.

See Solution

Problem 618

Find the limit: $\lim _{x \rightarrow 7} \frac{x^{3}-343}{x-7}$ .

See Solution

Problem 619

Find the limit of $\frac{f(x+h)-f(x)}{h}$ as $h \to 0$ for $f(x) = 2x + 3$ .

See Solution

Problem 620

Find the limit: $\lim _{x \rightarrow \infty} \frac{(5-x^{2})(7 x^{2}-2)}{-8 x^{3}(x^{2}-10)}$ . State DNE if infinite.

See Solution

Problem 621

Find the limit: $\lim _{x \rightarrow \infty} \frac{-24 x^{3}-9 x^{2}}{2 x^{3}-x^{2}-2 x+1}$ . State if it DNE.

See Solution

Problem 622

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{(1-7 x^{3})(x-8)}{(3 x^{2}-1)(2 x^{2}+3)}$ . State if it does not exist (DNE).

See Solution

Problem 623

Find the limit: $\lim _{x \rightarrow \infty} \frac{(1-7 x^{3})(x-8)}{(3 x^{2}-1)(2 x^{2}+3)}$ . State DNE if infinite.

See Solution

Problem 624

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{(1+10 x^{3})(4+x^{2})}{(10-x^{3})(x^{2}-3)}$ . State DNE if infinite.

See Solution

Problem 625

Find the limit: $\lim _{x \rightarrow \infty} \frac{40 x^{5}+15 x^{2}}{18 x^{3}-27 x^{5}+4-6 x^{2}}$ . State if it is infinite (DNE).

See Solution

Problem 626

Find the limit: $\lim _{x \rightarrow \infty} \frac{(1+10 x^{3})(4+x^{2})}{(10-x^{3})(x^{2}-3)}$ . State if it DNE.

See Solution

Problem 627

Find the limit: $\lim _{x \rightarrow \infty} \frac{(7+5 x^{2})(4 x^{3}-7)}{(2 x^{3}+9)(1+9 x^{2})}$ . State if it DNE.

See Solution

Problem 628

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{-2 x^{2}+x^{3}-33 x}}{10 x^{2}+6}$ . State DNE if infinite.

See Solution

Problem 629

Find the limit: $\lim _{x \rightarrow \infty} \frac{36+36 x+5 x^{2}}{90 x^{6}-29 x^{3}+2}$ . State if it DNE.

See Solution

Problem 630

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{42 x^{2}+16 x^{10}}}{5 x+4 x^{4}+4 x^{2}}$ . State DNE if infinite.

See Solution

Problem 631

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{-2 x^{2}+x^{3}-33 x}}{10 x^{2}+6}$ . State if it DNE.

See Solution

Problem 632

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt{42 x^{2}+16 x^{10}}}{5 x+4 x^{4}+4 x^{2}}$ . State if it DNE.

See Solution

Problem 633

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}$ . If infinite, state DNE.

See Solution

Problem 634

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}$ . State if it DNE.

See Solution

Problem 635

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}$ . State DNE if infinite.

See Solution

Problem 636

Determine the limit: $\lim _{x \rightarrow \infty} \frac{-x^{33}-9 x^{20}}{x^{33}-5 \log _{5} x}$ . What does it equal?

See Solution

Problem 637

Determine the limit: $\lim _{x \rightarrow \infty} \frac{-6 x^{91}}{x^{19}+x^{91}}$ . What does it equal?

See Solution

Problem 638

Determine the limit: $\lim _{x \rightarrow \infty} \frac{-7 x^{46}}{\ln x 6 e^{x}}$ . What does it equal?

See Solution

Problem 639

Find the horizontal asymptotes of the function $f(x)=1+2 e^{\frac{1}{x}}$ .

See Solution

Problem 640

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{7 \sin (3 x)}{3 x}$ .

See Solution

Problem 641

Find the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{7 \sin (3 x)}{3 x}$ .

See Solution

Problem 642

Evaluate the limit: $\lim_{x \to 0} \frac{-3x + 1 - \cos(x)}{2x}$

See Solution

Problem 643

(b) $\lim _{x \rightarrow+\infty, x>0} \frac{1}{x}=0$ .

See Solution

Problem 644

(1) (1,0 ponto) Use a Regra de L'Hôspital para calcular os seguintes limites, se for necessário: (a) $\lim _{x \rightarrow+\infty} \frac{\ln x}{\sqrt[3]{x}}$ (b) $\lim _{x \rightarrow 0} \frac{x}{e^{x}+\cos x}$ (c) $\lim _{x \rightarrow 0^{+}} \operatorname{sen} x \ln x$

See Solution

Problem 645

$\sqrt{\circ} 6$ gokazamb $\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e$

See Solution

Problem 646

Use a Regra de L'Hôspital para calcular os seguintes limites, se for necessário: (c) $\lim _{x \rightarrow 0^{+}} \operatorname{sen} x \ln x$

See Solution

Problem 647

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{3 \sin ^{2}(x)}{4 x}$ .

See Solution

Problem 648

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{-4 \sin (x)}{7 x}$ .

See Solution

Problem 649

Find the horizontal asymptotes of the function $f(x)=1+2 e^{\frac{1}{x}}$ .

See Solution

Problem 650

Find the horizontal asymptotes of the function $f(x)=\frac{3 x^{4}-4 x^{5}}{5 x^{3}-x^{4}}$ .

See Solution

Problem 651

Find the horizontal asymptotes for the function $f(x)=2 e^{\frac{1}{x}}$ .

See Solution

Problem 652

$\lim _{x \rightarrow \infty}\left(\frac{2 x^{2}-3}{x^{2}-5 x+3}\right)$

See Solution

Problem 653

$\lim _{x \rightarrow 0} \frac{\sqrt{2}-\sqrt{1+\cos x}}{\sin ^{2} x}$

See Solution

Problem 654

\#2. (18 points) Evaluate the following limits: (2a) (6 points) $\lim _{x \rightarrow \infty} x \sin \left(\frac{1}{x}\right)$ .

See Solution

Problem 655

1. $\lim _{x \rightarrow-\infty} \frac{3+2^{x}-\infty}{4-5^{x}}$ is (A) $-\frac{2}{5}$ (B) 0 (C) $\frac{3}{4}$

See Solution

Problem 656

fonksiyon için $\delta$ , sadece $\epsilon$ sayısına bağlıdır yani $x=t$ den bağımsız olmaktadır. 3.11 Örnek $f:(0,1] \longrightarrow \mathbb{R}, f(x)=\frac{1}{x}$ fonksiyonunun sürekli ancak düzgün sürekli olmadığım gösteriniz.

See Solution

Problem 657

Evaluate the limit: $\lim _{x \rightarrow 1} \frac{\sin \left(x^{2}+1\right)}{\sqrt{x}(5-x)}$

See Solution

Problem 658

Find the value(s) of $x$ where the function $f(x)$ is discontinuous, defined as: $f(x) = \begin{cases} x+1 & \text{if } x<0 \\ 2 \cos (\pi x) & \text{if } 0 \leq x \leq 2 \\ 6-x^{2} & \text{if } x>2 \end{cases}$

See Solution

Problem 659

Evaluate $\lim _{x \rightarrow 0} f(x)$ given $2-\cos x \leq f(x) \leq x^{2}+1$ .

See Solution

Problem 660

The function $f$ is defined for all $x$ in the interval $4<x<6$ . Which of the following statements, if true, implies that $\lim _{x \rightarrow 5} f(x)=17$ ? A) There exists a function $g$ with $f(x) \leq g(x)$ for $4<x<6$ , and $\lim _{x \rightarrow 5} g(x)=17$ .
B There exists a function $g$ with $g(x) \leq f(x)$ for $4<x<6$ , and $\lim _{x \rightarrow 5} g(x)=17$ . (C) There exist functions $g$ and $h$ with $f(x) \leq g(x) \leq h(x)$ for $4<x<6$ , and $\lim _{x \rightarrow 5} g(x)=\lim _{x \rightarrow 5} h(x)=17$ . (D) There exist functions $g$ and $h$ with $g(x) \leq f(x) \leq h(x)$ for $4<x<6$ , and $\lim _{x \rightarrow 5} g(x)=\lim _{x \rightarrow 5} h(x)=17$ .

See Solution

Problem 661

Find the limit $L$ . $L=\lim _{x \rightarrow-5} \frac{2 x^{2}+14 x+20}{x+5}$

See Solution

Problem 662

Mail - JENIFFER MARIE - [-/2 Points] DETAILS MY NOTES LARCALCET8 2.R.043.
Use a graphing utility to graph the function and estimate the limit $L$ . Use a table to reinforce your conclusion. Then find the limit $L$ by analytic methods. $\begin{array}{l} \lim _{x \rightarrow 0} \frac{\sqrt{2 x+4}-2}{x} \\ L=\square \end{array}$
Show My Work (Required)

See Solution

Problem 663

Find the limit $L$ (if it exists). (If an answer does not exist, enter DNE.) $\begin{array}{l} \lim _{x \rightarrow 9^{-}} \frac{x-9}{x^{2}-81} \\ L=\square \end{array}$
If it does not exist, explain why. The limit does not exist at $x=9$ because the function approaches different values from the left and right side of 9. The limit does not exist at $x=9$ because the function value is undefined at $x=9$ . The limit does not exist at $x=9$ because the function does not approach $f(9)$ as $x$ approaches 9 . The limit does not exist at $x=9$ because the function is not continuous at any $x$ value. The limit exists at $x=9$ .

See Solution

Problem 664

Determine whether $f(x)$ approaches $\infty$ or $-\infty$ as $x$ approaches 4 from the left and from the right. $\begin{array}{r} f(x)=\frac{-1}{(x-4)^{2}} \\ \lim _{x \rightarrow 4^{-}} \frac{-1}{(x-4)^{2}}=\square \\ \lim _{x \rightarrow 4^{+}} \frac{-1}{(x-4)^{2}}=\square \end{array}$ $\square$ $\square$

See Solution

Problem 665

Question 9 Not yet answered fligg quartion
If $f(x)=\left\{\begin{array}{ll}\frac{x^{2}-x}{x^{2}-1} & \text { if } x>1 \\ 2 k-3 & \text { if } x \leq 1\end{array}\right.$ is continuous at $x=1$ , then $k=$ $\frac{1}{2}$ $\frac{3}{4}$ 1 $\frac{7}{4}$ $\frac{-7}{4}$

See Solution

Problem 666

Find the limit $L$ (if it exists). (If an answer does not exist, enter DNE.) $\begin{array}{l} \lim _{x \rightarrow 5^{-}} \frac{|x-5|}{x-5} \\ L=\square \end{array}$
If it does not exist, explain why. The limit does not exist at $x=5$ because the function is not continuous at any $x$ value. The limit does not exist at $x=5$ because the function approaches different values from the left and right side of 5 . The limit does not exist at $x=5$ because the function value is undefined at $x=5$ . The limit does not exist at $x=5$ because the function does not approach $f(5)$ as $x$ approaches 5 . The limit exists at $x=5$ .

See Solution

Problem 667

Find $\lim _{x \rightarrow 4} \frac{x^{2}-x-12}{x-4}$

See Solution

Problem 668

(B) > (D) 1>2
9. Find lim 1-e* x+0 In (2-e*) (A) 210 (B) 1 (C) 0 (D) The limit does not exist. function given by f(x)=2 graph of fat

See Solution

Problem 669

d) $\lim _{x \rightarrow-\infty} \frac{3 x^{3}-2 x^{2}+1}{x^{2}+1}$

See Solution

Problem 670

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (6 x)}{\sin (2 x)}$ .

See Solution

Problem 671

Find $\delta$ such that if $|x-1|<\delta$ , then $|f(x)-1|<0.2$ , given $f(1)=1$ , $f(1.1)=0.8$ , $f(1.2)=1.2$ .

See Solution

Problem 672

Find $\delta$ so that if $0<|x-3|<\delta$ , then $|f(x)-2|<0.5$ for the function $f$ at points $(2.6,1.5)$ and $(3.8,2.5)$ .

See Solution

Problem 673

Find $\delta$ such that if $|x-4|<\delta$ , then $|\sqrt{x}-2|<0.4$ for $f(x)=\sqrt{x}$ .

See Solution

Problem 674

Find $\delta$ so that if $|x-1|<\delta$ , then $|f(x)-1|<0.2$ using points (.7,1.2) and (1.1,.8).

See Solution

Problem 675

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (4 x)}{2 x}=$

See Solution

Problem 676

Find $\delta$ so that if $0<|x-3|<\delta$ , then $|f(x)-2|<0.5$ using the graph of $f$ .

See Solution

Problem 677

Identify which functions are continuous for all real numbers: I. $f(x)=x^{1 / 3}$ II. $g(x)=\sec x$ III. $h(x)=e^{-x}$ (A) I only (B) I and II only (C) I and III only (D) I, II, and III

See Solution

Problem 678

Use the graph of $f(x)$ to answer the questions.
State the $x$ -values at which $f(x)^{5}$ is discontinuous. Note : List all $x$ -values and separate multiple $\square$

See Solution

Problem 679

5. $\lim _{x \rightarrow 2^{-}} \frac{x}{2-x}$

See Solution

Problem 680

$f(x)=\left\{\begin{array}{l} \frac{\sin (3 x)}{\sin (5 x)} \text{ if } x \neq 0 \\ \frac{a}{b} \text{ if } x=0 \end{array}\right.$
$g(x)=\left(x^{3}+2 x+1\right) \ln (x)$
المطلوب: إيجاد مجموعة تعريف الدوال $f(x)$ و $g(x)$ .

See Solution

Problem 681

Suppose that $\lim _{(x, y) \rightarrow(5,2)} f(x, y)=7$ . What can you say about the value of $f(5,2)$ ? We can say $f(5,2)=\infty$ . We can say $f(5,2)$ is an open point. We can say $\lim _{(x, y) \rightarrow(5,2)} f(x, y)=f(5,2)=7$ . We can say $\underset{(x, y) \rightarrow(5,2)}{ } f(x, y)=f(5,2)=\infty$ . In general, nothing can be said about the value of $f(5,2)$ .
What if $f$ is continuous? We can say $f(5,2)=\infty$ . We can say $f(5,2)$ is an open point. We can say $(x, y) \rightarrow(5,2)$ . We can say $\lim _{(x, y) \rightarrow(5,2)} f(x, y)=f(5,2)=\infty$ . In general, nothing can be said about the value of $f(5,2)$ .

See Solution

Problem 682

Determine the set of points at which the function is continuous. $f(x, y)=\left\{\begin{array}{ll} \frac{x^{2} y^{3}}{2 x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 1 & \text { if }(x, y)=(0,0) \end{array}\right.$ $\{(x, y) \mid x>0$ and $y>0\}$ $\{(x, y) \mid(x, y) \neq(0,0)\}$ $\{(x, y) \mid x \in \mathbb{R}$ and $y \neq 0\}$ $\{(x, y) \mid x \in \mathbb{R}$ and $y \in \mathbb{R}\}$ $\{(x, y) \mid x \cdot y \neq 0\}$

See Solution

Problem 683

7. Evaluate $\lim _{x \rightarrow 0}(2-x) \tan \left(\frac{\pi x}{2}\right)$ $\begin{array}{l} \lim _{x \rightarrow 0}(-2-x)^{\tan \left(\frac{\pi x}{2}\right)} \\ =(2-0)^{\tan \frac{\pi(0)}{2}} \\ =(2-0)^{\tan \frac{0}{2}} \\ =(2-0)^{\circ} \\ =1 \end{array}$

See Solution

Problem 684

Exercice 1. 1) Montrer en utilisant la définition que $\lim _{x \rightarrow 1} 4 x+2=6, \lim _{x \rightarrow 0} x^{3}+x+2=2, \lim _{x \rightarrow+\infty} \frac{1}{x^{3}+1}=0$ 2) Montrer que les limites $\lim _{x \rightarrow 1} E(x), \lim _{x \rightarrow 0} \cos \frac{1}{x}$ n'existent pas.
Exercice 2. Calculer les limites suivantes:
1. $\lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x^{3}}$ ,
2. $\lim _{x \rightarrow \frac{\pi}{2}} \frac{e^{\cos x}-1}{x-\frac{\pi}{2}}$ ,
3. $\lim _{x \rightarrow 0} \frac{x}{a} E$ $-1-2 x)$ ,
5. $\lim _{x \rightarrow+\infty}\left(\sqrt{x^{2}+2 x-1}-2 x\right)$ , 6. $\lim _{x \rightarrow+\infty}\left(1+\frac{1}{x}\right)^{x}$ ,
7. $\lim _{x \rightarrow-\infty}\left(\frac{x-1}{x+1}\right)^{x}$ .
Exercice 3. Soit $f: \mathbb{R} \longrightarrow \mathbb{R}$ , une fonction définie par $f(x)=\left\{\begin{array}{ll} x & \text { si }|x| \leqslant 1 \\ x^{2}+a x+b & \text { si }|x|>1 \end{array}\right.$
Où $a, b \in \mathbb{R}$ . Déterminer les valeurs de $a$ et $b$ pour que $f$ soit continue sur $\mathbb{R}$ . Exercice 4. Soit $f: \mathbb{R} \longrightarrow \mathbb{R}$ la fonction définie par $f(x)=\left\{\begin{array}{l} 1 \text { si } x \in \mathbb{Q} \\ 0 \text { si } \notin \mathbb{Q} . \end{array}\right.$
Montrer que $f$ est discontinue en tout point de $\mathbb{R}$ . Exercice 5. Soit $f$ une fonction définie par $f:[a, b] \rightarrow[a, b]$ , telle que $\left|f\left(x_{1}\right)-f\left(x_{2}\right)\right|<\left|x_{1}-x_{2}\right|, \quad \forall x_{1}, x_{2} \in[a, b], \quad\left(x_{1} \neq x_{2}\right) .$ 1) Montrer que $f$ est continue sur $[a, b]$ . 2) Montrer que l'équation $f(x)=x$ admet une solution unique.
Exercice 6. Étudier dans chacun des cas suivants si la fonction $f$ est prolongeable par continuité sur $\mathbb{R}$ . 1) $f: x \longrightarrow \frac{1-\cos \sqrt{|x|}}{|x|}$ , 2) $f: x \longrightarrow \sin (x) \cdot \sin \left(\frac{1}{x}\right)$ , 3) $f: x \longrightarrow \frac{x^{3}+1}{x^{2}+3 x+2}$ , 4) $f: x \longrightarrow 1-\cos \left(1-\cos \frac{1}{x-2}\right)$ , 5) $f: x \longrightarrow \frac{\sin |x-4|}{x-4}$ , 6) $f: x \longrightarrow \frac{1}{1-x}-\frac{2}{1-x^{2}}$ .

See Solution

Problem 685

8. Sketch a graph where $f(4)$ does not exist but $\lim _{x \rightarrow 4} \quad f(x)$ does exist.

See Solution

Problem 686

For each of the following forms determine whether the following limit type is indeterminate, always has a fixed finite value, or never has a fixed finite value. In the first case answer IND, in the second case enter the numerical value, and in the third case answer DNE. To discourage blind guessing, this problem is graded on the following scale $0-9$ correct $=0$ $10-13$ correct $=.3$ 14-16 correct $=.5$ 17-19 correct $=.7$ Note that l'Hospital's rule (in some form) may ONLY be applied to indeterminate forms. $\square$ 1. $1 \cdot \infty$
2. $\pi^{-\infty}$
3. $\infty \cdot \infty$
4. $\infty^{1}$
5. $\infty^{-\infty}$
6. $\pi^{\infty}$
7. $0^{-\infty}$
8. $\infty^{-e}$
9. $\infty^{\infty}$
10. $1^{0}$
11. $\frac{1}{-\infty}$ $12.1^{\infty}$
13. $\frac{0}{\infty}$
14. $0 \cdot \infty$
15. $0^{\infty}$
16. $\frac{\infty}{0}$ $17.0^{0}$ $18.1^{-\infty}$

See Solution

Problem 687

(13) $\log _{\mathrm{b}} \mathrm{a} \times \log _{\mathrm{c}} \mathrm{b} \times \log _{\mathrm{a}} \mathrm{c}=$ (a) zero (b) 1 (c) a b c (d) ac (14) $\operatorname{Lim}_{x \rightarrow 2}\left(3 a^{2}\right)=$ $\qquad$ (a) 3 (b) 12 (c) $3 a^{2}$ (d) 6 (15) $\operatorname{Lim}_{x \rightarrow 0} \frac{x^{2}-x}{x}=$ $\qquad$ (a) zero (b) -1 (c) $1 \cdot$ (d) Doesn't exist. (16) $\operatorname{Lim}_{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x}=$ $\qquad$ (a) zero (b) $\sqrt{2}$ (c) $\frac{1}{2}$ (d) as no existence (17) If $\operatorname{Lim}_{x \rightarrow 2} \frac{x^{2}-4 a}{x-2}$ exists, then $\mathrm{a}=$ $\qquad$ (a) -1 (b) 1 (c) 2 (d) 4 (18) $\operatorname{Lim}_{x \rightarrow 2} \frac{x^{5}-32}{x^{3}-8}=$ $\qquad$ (a) 4 (b) $\frac{5}{3}$ (c) zero (d) $6 \frac{2}{3}$

See Solution

Problem 688

(10) $\lim \sin (x)$

See Solution

Problem 689

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos x}{3 \sin ^{2} x}$ .

See Solution

Problem 690

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan (2 x)}{3 x}=$ (A) $\frac{1}{3}$ (B) $\frac{1}{2}$ (C) $\frac{2}{3}$ (D) 2

See Solution

Problem 691

Find the limit: $\lim _{x \rightarrow \infty} \frac{8 x^{2}-5}{5 x^{2}+x-3}$ .

See Solution

Problem 692

Find the limit: $\lim _{x \rightarrow-\infty} \frac{6 x^{5}-x}{x^{4}+3}$ .

See Solution

Problem 693

78. If $f(x)=\ln (x)$ and $g$ is a differentiable function with domain $x>0$ such that $\lim _{x \rightarrow \infty} g(x)=\infty$ and $g^{\prime}$ has a horizontal asymptote at $y=4$ then $\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}$ is A. 0 B. -4 C. 4 D. nonexistent

See Solution

Problem 694

23-28 True-False Determine whether tr - statement is true or false. Explain your answer.
23. If $f(x)$ is continuous at $x=c$ , then so is $|f(x)|$ .
24. If $|f(x)|$ is continuous at $x=c$ , then so is $f(x)$ .
25. If $f$ and $g$ are discontinuous at $x=c$ , then so is $f+g$ .
26. If $f$ and $g$ are discontinuous at $x=c$ , then so is $f g$ .

See Solution

Problem 695

$\lim_{x \rightarrow 8^{-}} \frac{3p(x) - 2x}{15 - 3x} = ?$
Given that as $x$ approaches 8 from the left, the value of $p(x)$ approaches 2, and as $x$ approaches 8 from the right, the value of $p(x)$ approaches 3.

See Solution

Problem 696

(1 point)
Evaluate the limit $\lim _{x \rightarrow \infty} \sqrt{x^{2}+5 x+6}-x$

See Solution

Problem 697

Find the limit: $\lim _{x \rightarrow 0}\left(\frac{4}{x}-\frac{4}{\sin (x)}\right)=$ $\square$ (Enter undefined if the limit does not exist.)
Preview My Answers Submit Answers

See Solution

Problem 698

Evaluate the limit using L'Hospital's rule if necessary. $\lim _{x \rightarrow 0^{+}} x^{9 \sin (x)}$
Answer: $\square$

See Solution

Problem 699

(1 point)
Evaluate the following limits, using L'Hopital's rule if appropriate. $\lim _{x \rightarrow 0^{+}} \sqrt[4]{x} \ln (x)=$ $\square$ Preview My Answers Submit Answers

See Solution

Problem 700

Find the limit. Use l'Hospital's Rule where appropriate. $\lim _{x \rightarrow-\infty} x^{2} e^{x}$
Limit: $\square$
Preview My Answers Submit Answers
You have attempted this problem 0 times.

See Solution

Limits & Continuity

Problem 601

Find lim⁡x→+∞2x+5x2−7x+3\lim _{x \rightarrow+\infty} \frac{2 x+5}{x^{2}-7 x+3}limx→+∞​x2−7x+32x+5​Add your answer Integer, decimal, or E notation allowed

Problem 602

Find lim⁡x→4x2−x−12x−4\lim _{x \rightarrow 4} \frac{x^{2}-x-12}{x-4}limx→4​x−4x2−x−12​Add your answer Integer, decimal, or E notation allowed

Problem 603

Determine where f(x)\mathrm{f}(\mathrm{x})f(x) is continuous. f(x)=sin⁡−12xf(x)=\sin ^{-1} 2 xf(x)=sin−12x (A) −1/2-1 / 2−1/2 and 1/21 / 21/2 (B) 1 and - 1/21 / 21/2 (C) -1 and 0 (D) -1 and -3

Problem 604

Is the function f such that f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ for x>0\mathrm{x}>0x>0 and f(0)=0\mathrm{f}(0)=0f(0)=0 continuous over [0,1][0,1][0,1] ? (A) no (B) yes (C) maybe (D) none of the answer

Problem 605

Evaluate the limit: lim⁡x→04x+49−7x=\lim _{x \rightarrow 0} \frac{\sqrt{4 x+49}-7}{x}=x→0lim​x4x+49​−7​= □\square□

Problem 606

Problem 607

Problem 608

ابحث في اتصال ق ( س) عند س = 1 ، وغند س = ب من جهة اليسار وايضا ابحث الاتصال على الفترة ( . ، )

Problem 609

Problem 610

lim⁡[x→∞](x2+3x+2+x+1)\lim [x \rightarrow \infty]\left(\sqrt{x^{2}+3 x+2}+x+1\right)lim[x→∞](x2+3x+2​+x+1)

Problem 611

lim⁡x→0[x−7(sin⁡x−x5120+x36−x)]\lim _{x \rightarrow 0}\left[x^{-7}\left(\sin x-\frac{x^{5}}{120}+\frac{x^{3}}{6}-x\right)\right]limx→0​[x−7(sinx−120x5​+6x3​−x)]

Problem 612

4.

Problem 613

5) lim⁡x→−∞−16xx2+16\lim _{x \rightarrow-\infty}-\frac{16 x}{x^{2}+16}limx→−∞​−x2+1616x​

Problem 614

lim⁡x→+∞1+x−xln⁡x\lim _{x \rightarrow+\infty} 1+x-x \ln xlimx→+∞​1+x−xlnx

Problem 615

Find the secant line slope for f(x)=x3−11f(x)=\sqrt[3]{x}-11f(x)=3x​−11 at x1=750x_1=750x1​=750 and x2=6x_2=6x2​=6.

Problem 616

Find the limit: lim⁡x→0(cot⁡(6x)⋅sin⁡(2x))\lim _{x \rightarrow 0}(\cot (6 x) \cdot \sin (2 x))limx→0​(cot(6x)⋅sin(2x)).

Problem 617

Find the left-hand and right-hand limits of f(x)=sin⁡(x)f(x) = \sin(x)f(x)=sin(x) as xxx approaches 1, and the limit as xxx approaches -1.

Problem 618

Find the limit: lim⁡x→7x3−343x−7\lim _{x \rightarrow 7} \frac{x^{3}-343}{x-7}limx→7​x−7x3−343​.

Problem 619

Find the limit of f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ as h→0h \to 0h→0 for f(x)=2x+3f(x) = 2x + 3f(x)=2x+3.

Problem 620

Find the limit: lim⁡x→∞(5−x2)(7x2−2)−8x3(x2−10)\lim _{x \rightarrow \infty} \frac{(5-x^{2})(7 x^{2}-2)}{-8 x^{3}(x^{2}-10)}x→∞lim​−8x3(x2−10)(5−x2)(7x2−2)​. State DNE if infinite.

Problem 621

Find the limit: lim⁡x→∞−24x3−9x22x3−x2−2x+1\lim _{x \rightarrow \infty} \frac{-24 x^{3}-9 x^{2}}{2 x^{3}-x^{2}-2 x+1}x→∞lim​2x3−x2−2x+1−24x3−9x2​. State if it DNE.

Problem 622

Find the limit as xxx approaches infinity: lim⁡x→∞(1−7x3)(x−8)(3x2−1)(2x2+3)\lim _{x \rightarrow \infty} \frac{(1-7 x^{3})(x-8)}{(3 x^{2}-1)(2 x^{2}+3)}x→∞lim​(3x2−1)(2x2+3)(1−7x3)(x−8)​. State if it does not exist (DNE).

Problem 623

Find the limit: lim⁡x→∞(1−7x3)(x−8)(3x2−1)(2x2+3)\lim _{x \rightarrow \infty} \frac{(1-7 x^{3})(x-8)}{(3 x^{2}-1)(2 x^{2}+3)}x→∞lim​(3x2−1)(2x2+3)(1−7x3)(x−8)​. State DNE if infinite.

Problem 624

Find the limit as xxx approaches infinity: lim⁡x→∞(1+10x3)(4+x2)(10−x3)(x2−3)\lim _{x \rightarrow \infty} \frac{(1+10 x^{3})(4+x^{2})}{(10-x^{3})(x^{2}-3)}x→∞lim​(10−x3)(x2−3)(1+10x3)(4+x2)​. State DNE if infinite.

Problem 625

Find the limit: lim⁡x→∞40x5+15x218x3−27x5+4−6x2\lim _{x \rightarrow \infty} \frac{40 x^{5}+15 x^{2}}{18 x^{3}-27 x^{5}+4-6 x^{2}}x→∞lim​18x3−27x5+4−6x240x5+15x2​. State if it is infinite (DNE).

Problem 626

Find the limit: lim⁡x→∞(1+10x3)(4+x2)(10−x3)(x2−3)\lim _{x \rightarrow \infty} \frac{(1+10 x^{3})(4+x^{2})}{(10-x^{3})(x^{2}-3)}x→∞lim​(10−x3)(x2−3)(1+10x3)(4+x2)​. State if it DNE.

Problem 627

Find the limit: lim⁡x→∞(7+5x2)(4x3−7)(2x3+9)(1+9x2)\lim _{x \rightarrow \infty} \frac{(7+5 x^{2})(4 x^{3}-7)}{(2 x^{3}+9)(1+9 x^{2})}x→∞lim​(2x3+9)(1+9x2)(7+5x2)(4x3−7)​. State if it DNE.

Problem 628

Find the limit: lim⁡x→∞−2x2+x3−33x10x2+6\lim _{x \rightarrow \infty} \frac{\sqrt{-2 x^{2}+x^{3}-33 x}}{10 x^{2}+6}x→∞lim​10x2+6−2x2+x3−33x​​. State DNE if infinite.

Problem 629

Find the limit: lim⁡x→∞36+36x+5x290x6−29x3+2\lim _{x \rightarrow \infty} \frac{36+36 x+5 x^{2}}{90 x^{6}-29 x^{3}+2}x→∞lim​90x6−29x3+236+36x+5x2​. State if it DNE.

Problem 630

Find the limit: lim⁡x→∞42x2+16x105x+4x4+4x2\lim _{x \rightarrow \infty} \frac{\sqrt{42 x^{2}+16 x^{10}}}{5 x+4 x^{4}+4 x^{2}}x→∞lim​5x+4x4+4x242x2+16x10​​. State DNE if infinite.

Problem 631

Find the limit: lim⁡x→∞−2x2+x3−33x10x2+6\lim _{x \rightarrow \infty} \frac{\sqrt{-2 x^{2}+x^{3}-33 x}}{10 x^{2}+6}x→∞lim​10x2+6−2x2+x3−33x​​. State if it DNE.

Problem 632

Find the limit as xxx approaches infinity: lim⁡x→∞42x2+16x105x+4x4+4x2\lim _{x \rightarrow \infty} \frac{\sqrt{42 x^{2}+16 x^{10}}}{5 x+4 x^{4}+4 x^{2}}x→∞lim​5x+4x4+4x242x2+16x10​​. State if it DNE.

Problem 633

Find the limit as xxx approaches infinity: lim⁡x→∞−5x4+33x5−64x7310x+8+9x3\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}x→∞lim​10x+8+9x33−5x4+33x5−64x7​​. If infinite, state DNE.

Problem 634

Find the limit: lim⁡x→∞−5x4+33x5−64x7310x+8+9x3\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}x→∞lim​10x+8+9x33−5x4+33x5−64x7​​. State if it DNE.

Problem 635

Find the limit: lim⁡x→∞−5x4+33x5−64x7310x+8+9x3\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}x→∞lim​10x+8+9x33−5x4+33x5−64x7​​. State DNE if infinite.

Problem 636

Determine the limit: lim⁡x→∞−x33−9x20x33−5log⁡5x\lim _{x \rightarrow \infty} \frac{-x^{33}-9 x^{20}}{x^{33}-5 \log _{5} x}x→∞lim​x33−5log5​x−x33−9x20​. What does it equal?

Problem 637

Determine the limit: lim⁡x→∞−6x91x19+x91\lim _{x \rightarrow \infty} \frac{-6 x^{91}}{x^{19}+x^{91}}x→∞lim​x19+x91−6x91​. What does it equal?

Problem 638

Determine the limit: lim⁡x→∞−7x46ln⁡x6ex\lim _{x \rightarrow \infty} \frac{-7 x^{46}}{\ln x 6 e^{x}}x→∞lim​lnx6ex−7x46​. What does it equal?

Problem 639

Find the horizontal asymptotes of the function f(x)=1+2e1xf(x)=1+2 e^{\frac{1}{x}}f(x)=1+2ex1​.

Problem 640

Evaluate the limit: lim⁡x→07sin⁡(3x)3x\lim _{x \rightarrow 0} \frac{7 \sin (3 x)}{3 x}limx→0​3x7sin(3x)​.

Problem 641

Find the limit as xxx approaches 0: lim⁡x→07sin⁡(3x)3x\lim _{x \rightarrow 0} \frac{7 \sin (3 x)}{3 x}limx→0​3x7sin(3x)​.

Find $\lim _{x \rightarrow+\infty} \frac{2 x+5}{x^{2}-7 x+3}$
Add your answer Integer, decimal, or E notation allowed

Find $\lim _{x \rightarrow 4} \frac{x^{2}-x-12}{x-4}$
Add your answer Integer, decimal, or E notation allowed

Determine where $\mathrm{f}(\mathrm{x})$ is continuous. $f(x)=\sin ^{-1} 2 x$ (A) $-1 / 2$ and $1 / 2$ (B) 1 and - $1 / 2$ (C) -1 and 0 (D) -1 and -3

Is the function f such that $f(x)=\frac{1}{x}$ for $\mathrm{x}>0$ and $\mathrm{f}(0)=0$ continuous over $[0,1]$ ? (A) no (B) yes (C) maybe (D) none of the answer

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{4 x+49}-7}{x}=$ $\square$

$\lim [x \rightarrow \infty]\left(\sqrt{x^{2}+3 x+2}+x+1\right)$

$\lim _{x \rightarrow 0}\left[x^{-7}\left(\sin x-\frac{x^{5}}{120}+\frac{x^{3}}{6}-x\right)\right]$

5) $\lim _{x \rightarrow-\infty}-\frac{16 x}{x^{2}+16}$

$\lim _{x \rightarrow+\infty} 1+x-x \ln x$

Find the secant line slope for $f(x)=\sqrt[3]{x}-11$ at $x_1=750$ and $x_2=6$ .

Find the limit: $\lim _{x \rightarrow 0}(\cot (6 x) \cdot \sin (2 x))$ .

Find the left-hand and right-hand limits of $f(x) = \sin(x)$ as $x$ approaches 1, and the limit as $x$ approaches -1.

Find the limit: $\lim _{x \rightarrow 7} \frac{x^{3}-343}{x-7}$ .

Find the limit of $\frac{f(x+h)-f(x)}{h}$ as $h \to 0$ for $f(x) = 2x + 3$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{(5-x^{2})(7 x^{2}-2)}{-8 x^{3}(x^{2}-10)}$ . State DNE if infinite.

Find the limit: $\lim _{x \rightarrow \infty} \frac{-24 x^{3}-9 x^{2}}{2 x^{3}-x^{2}-2 x+1}$ . State if it DNE.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{(1-7 x^{3})(x-8)}{(3 x^{2}-1)(2 x^{2}+3)}$ . State if it does not exist (DNE).

Find the limit: $\lim _{x \rightarrow \infty} \frac{(1-7 x^{3})(x-8)}{(3 x^{2}-1)(2 x^{2}+3)}$ . State DNE if infinite.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{(1+10 x^{3})(4+x^{2})}{(10-x^{3})(x^{2}-3)}$ . State DNE if infinite.

Find the limit: $\lim _{x \rightarrow \infty} \frac{40 x^{5}+15 x^{2}}{18 x^{3}-27 x^{5}+4-6 x^{2}}$ . State if it is infinite (DNE).

Find the limit: $\lim _{x \rightarrow \infty} \frac{(1+10 x^{3})(4+x^{2})}{(10-x^{3})(x^{2}-3)}$ . State if it DNE.

Find the limit: $\lim _{x \rightarrow \infty} \frac{(7+5 x^{2})(4 x^{3}-7)}{(2 x^{3}+9)(1+9 x^{2})}$ . State if it DNE.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{-2 x^{2}+x^{3}-33 x}}{10 x^{2}+6}$ . State DNE if infinite.

Find the limit: $\lim _{x \rightarrow \infty} \frac{36+36 x+5 x^{2}}{90 x^{6}-29 x^{3}+2}$ . State if it DNE.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{42 x^{2}+16 x^{10}}}{5 x+4 x^{4}+4 x^{2}}$ . State DNE if infinite.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{-2 x^{2}+x^{3}-33 x}}{10 x^{2}+6}$ . State if it DNE.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt{42 x^{2}+16 x^{10}}}{5 x+4 x^{4}+4 x^{2}}$ . State if it DNE.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}$ . If infinite, state DNE.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}$ . State if it DNE.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-5 x^{4}+33 x^{5}-64 x^{7}}}{10 x+8+9 x^{3}}$ . State DNE if infinite.

Determine the limit: $\lim _{x \rightarrow \infty} \frac{-x^{33}-9 x^{20}}{x^{33}-5 \log _{5} x}$ . What does it equal?

Determine the limit: $\lim _{x \rightarrow \infty} \frac{-6 x^{91}}{x^{19}+x^{91}}$ . What does it equal?

Determine the limit: $\lim _{x \rightarrow \infty} \frac{-7 x^{46}}{\ln x 6 e^{x}}$ . What does it equal?

Find the horizontal asymptotes of the function $f(x)=1+2 e^{\frac{1}{x}}$ .

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{7 \sin (3 x)}{3 x}$ .

Find the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{7 \sin (3 x)}{3 x}$ .

Evaluate the limit: $\lim_{x \to 0} \frac{-3x + 1 - \cos(x)}{2x}$

(b) $\lim _{x \rightarrow+\infty, x>0} \frac{1}{x}=0$ .

$\sqrt{\circ} 6$ gokazamb $\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e$

Use a Regra de L'Hôspital para calcular os seguintes limites, se for necessário: (c) $\lim _{x \rightarrow 0^{+}} \operatorname{sen} x \ln x$

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{3 \sin ^{2}(x)}{4 x}$ .

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{-4 \sin (x)}{7 x}$ .

Find the horizontal asymptotes of the function $f(x)=1+2 e^{\frac{1}{x}}$ .

Find the horizontal asymptotes of the function $f(x)=\frac{3 x^{4}-4 x^{5}}{5 x^{3}-x^{4}}$ .

Find the horizontal asymptotes for the function $f(x)=2 e^{\frac{1}{x}}$ .

$\lim _{x \rightarrow \infty}\left(\frac{2 x^{2}-3}{x^{2}-5 x+3}\right)$

$\lim _{x \rightarrow 0} \frac{\sqrt{2}-\sqrt{1+\cos x}}{\sin ^{2} x}$

\#2. (18 points) Evaluate the following limits: (2a) (6 points) $\lim _{x \rightarrow \infty} x \sin \left(\frac{1}{x}\right)$ .

1. $\lim _{x \rightarrow-\infty} \frac{3+2^{x}-\infty}{4-5^{x}}$ is (A) $-\frac{2}{5}$ (B) 0 (C) $\frac{3}{4}$

fonksiyon için $\delta$ , sadece $\epsilon$ sayısına bağlıdır yani $x=t$ den bağımsız olmaktadır. 3.11 Örnek $f:(0,1] \longrightarrow \mathbb{R}, f(x)=\frac{1}{x}$ fonksiyonunun sürekli ancak düzgün sürekli olmadığım gösteriniz.

Evaluate the limit: $\lim _{x \rightarrow 1} \frac{\sin \left(x^{2}+1\right)}{\sqrt{x}(5-x)}$

Evaluate $\lim _{x \rightarrow 0} f(x)$ given $2-\cos x \leq f(x) \leq x^{2}+1$ .

Find the limit $L$ . $L=\lim _{x \rightarrow-5} \frac{2 x^{2}+14 x+20}{x+5}$

Find $\lim _{x \rightarrow 4} \frac{x^{2}-x-12}{x-4}$

(B) > (D) 1>2
9. Find lim 1-e* x+0 In (2-e*) (A) 210 (B) 1 (C) 0 (D) The limit does not exist. function given by f(x)=2 graph of fat

d) $\lim _{x \rightarrow-\infty} \frac{3 x^{3}-2 x^{2}+1}{x^{2}+1}$

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (6 x)}{\sin (2 x)}$ .

Find $\delta$ such that if $|x-1|<\delta$ , then $|f(x)-1|<0.2$ , given $f(1)=1$ , $f(1.1)=0.8$ , $f(1.2)=1.2$ .

Find $\delta$ so that if $0<|x-3|<\delta$ , then $|f(x)-2|<0.5$ for the function $f$ at points $(2.6,1.5)$ and $(3.8,2.5)$ .

Find $\delta$ such that if $|x-4|<\delta$ , then $|\sqrt{x}-2|<0.4$ for $f(x)=\sqrt{x}$ .

Find $\delta$ so that if $|x-1|<\delta$ , then $|f(x)-1|<0.2$ using points (.7,1.2) and (1.1,.8).