Browse Limits & Continuity problems Studdy Solved!

Problem 501

Use the graph of the function $f(x)$ shown below in order to determine all values of $a$ for which $\lim_{x \to a} f(x) = -3$ on the interval $-9 < x < 9$ .
Answer Attempt 2 out of 2
Additional Solution No Solution
$a =$

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Problem 502

Over which interval does function $f(x) = \frac{1}{\sqrt{x+5}}$ have a point (points) of discontinuity?
(A) ]-5, + $\infty$ [ (B) ]- $\infty$ , -5] (C) ]0, -5[ (D) ]0, -5[ (E) [-5, + $\infty$ [

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Problem 503

Problem 1. (1 point)
A function $f$ and value $a$ are given. Approximate the limit of the difference quotient, $\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ , using $h= \pm 0.1, \pm 0.01$ . $f(x)=\sin x, \quad a=\pi$
When $h=0.1, \frac{f(a+h)-f(a)}{h}=$ $\square$ When $h=-0.1, \frac{f(a+h)-f(a)}{h}=$ $\square$ When $h=0.01, \frac{f(a+h)-f(a)}{h}=$ $\square$ When $h=-0.01, \frac{f(a+h)-f(a)}{h}=$ $\square$ Note: You can earn partial credit on this problem.

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Problem 504

Estimate the soccer ball's instantaneous velocity at $t=2$ s using the average velocities given. Round to two decimal places.

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Problem 505

Prove that $\lim _{x \rightarrow 6} 4 x=24$ using the precise definition of a limit.

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Problem 506

Find the average velocity $\bar{v}$ of a stone with height $h(t)=30t-4.9t^2$ over intervals around $t=3$ .

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Problem 507

Evaluate the function $f(x)$ and determine which statements about its limit and continuity at $x=1$ are true. Options: A. Only I, B. Only II, C. I and II, D. None, E. All.

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Problem 508

Find the average velocity $\bar{v}$ of a stone tossed with $h(t)=35t-4.9t^2$ over intervals $[1,1.01], [1,1.001], [1,1.0001], [0.9999,1], [0.999,1], [0.99,1]$ . Round to three decimal places.

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Problem 509

21. $\lim _{x \rightarrow \infty} \frac{4-\sqrt{x}}{2+\sqrt{x}}$
23. $\lim _{x \rightarrow \infty} \frac{\sqrt{x+3 x^{2}}}{4 x-1}$

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Problem 510

Let $f(x) = \begin{cases} ax & \text{if } x \le 1, \\ bx^2 + x + 1 & \text{if } x > 1. \end{cases}$
a. Find all combinations of $a$ and $b$ such that $f$ is continuous at $x = 1$ . Write your answer as a formula for $a$ in terms of $b$ . $a =$

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Problem 511

Suppose $C$ is a constant and $g(x)$ is a function of $x$ such that $C x+2 \leq g(x) \leq x-4$ for all values of $x$ near 16 but not equal to 16 . We wish to find $\lim _{x \rightarrow 16} g(x)$ by using the Squeeze Theorem. a. In order for the Squeeze Theorem to be applicable in this case, what must the value of $C$ be equal to? Enter your answer as an exact value (enter as a fraction if necessary). $C=\square$ b. Find $\lim _{x \rightarrow 16} g(x)$ using the Squeeze Theorem. Enter your answer as an exact value (enter as a fraction if necessary).
Answer: $\square$

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Problem 512

$20. \lim_{x \to \frac{\pi}{2}} \ln{x} \tan{(\frac{x}{2})}$

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Problem 513

Problem 2. (1 point) Using: $\lim _{x \rightarrow 7} f(x)=6$ and $\lim _{x \rightarrow 7} g(x)=4$ , evaluate $\lim _{x \rightarrow 7} \frac{f(x)+g(x)}{8 f(x)}$
Limit $=$ $\square$ Enter DNE if the limit does not exist.

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Problem 514

$3.$ $h(x) = 1$ For what value of 'k' is $f(x)$ continuous on $(-\infty, \infty)?$ a) $f(x) = \begin{cases} x^2 - k^2 & , x < 4 \\ kx + 20 & , x \geq 4 \end{cases}$ $4^2 - k^2 = k(4) + 20$ $16 - k^2 = 4k + 20$ $-k^2 - 4k - 4$ $-k^2 - 2k - 2k - 4$ $-k(k+2) - 2(k+2)$ $(k+2)(-k-2)$ $k = -2$ b) $g(x) = \begin{cases} \frac{4x^2 + 9x - 9}{x+3} & , x \neq 3 \\ 4x + k & , x = 3 \end{cases}$ $\frac{4 \cdot 3^2 + 9 \cdot 3 - 9}{3+3}$ $\frac{36 + 18}{6} = 9$ $4 \cdot 3 + k$ $12 + k = 9$ $12 - 9 + k = 0$ $3 = k$
meaning both limits but the function $H(x)$ doesn't matter So not continuous

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Problem 515

Q5 (3 points) Let $f(x)=\frac{x}{x^{3}-x}$ (a) Determine whether $f$ is even, odd, or neither. (b) Give the equation(s) of horizontal asymptote(s). (c) Give the equation(s) of vertical asymptote(s).
Q6. (3 points) Find the point on the line $y=2 x-4$ that is closest to the point $(0,1)$ .

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Problem 516

$\lim _{x \rightarrow-3} \frac{4-\sqrt{x^{2}+7}}{3 x+9}$

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Problem 517

Find the limits: 1. $\lim _{x \rightarrow 3} \frac{3-x}{x^{2}-9}$ and 2. $\lim _{x \rightarrow 4} \frac{x^{2}-5 x+4}{x^{2}-2 x-8}$ .

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Problem 518

Find the limit as x approaches 1: $\lim_{{x \to 1}} \frac{\sqrt{2 x^{3}+x+1}-2 x}{x-1}$

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Problem 519

Find the limits: 1. $\lim _{x \rightarrow 4} \frac{\sqrt{x+5}-3}{x-4}$ , 2. $\lim _{x \rightarrow \frac{\pi}{4}} \frac{1-\tan x}{\sin x-\cos x}$ . Also, find $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ for $f(x)=x^{2}-x$ and $f(x)=\frac{1}{x+3}$ .

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Problem 520

Find the limit as $h$ approaches 0 for the difference quotient of: 1. $f(x)=x^{2}-x$ , 2. $f(x)=\frac{1}{x+3}$ .

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Problem 521

מה הגבול של $\left(\sqrt[3]{x^{3}+x^{2}}-x\right)$ כש- $x$ מתקרב למינוס אינסוף?

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Problem 522

Evaluate the piecewise function $f$ at $x=3$ to check if it's continuous and differentiable. What is true about $f$ at $x=3$ ?

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Problem 523

Given that $f(x)$ is continuous at $x=-9$ with $f(-9)=3$ and $f^{\prime}(-9)=-1$ , which statements are true?
1. $f(x)$ is differentiable at $x=-9$
2. $\lim _{x \rightarrow 3} f(x)=-9$
3. $\lim _{x \rightarrow -9} f(x)=3$
4. None of the above.

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Problem 524

Given $g(x)$ is differentiable at $x=-3$ with $g(-3)=9$ and $g'(-3)=0$ , which statements are true?

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Problem 525

Given the piecewise function $f$ , determine if it is continuous or differentiable at $x=-3$ . Options: 1) neither, 2) continuous only, 3) both.

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Problem 526

Is the function $f(x)=\left\{\begin{array}{cc}\sin (x) & x<\frac{\pi}{2} \\ \frac{2 x}{\pi} & x \geq \frac{\pi}{2}\end{array}\right.$ continuous at $x=\frac{\pi}{2}$ ? Explain.

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Problem 527

Find the acceleration due to gravity at 6989 m, the horizontal asymptote of $g(h)$ , and solve $g(h)=0$ .

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Problem 528

When can average speed equal instantaneous speed?

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Problem 529

True or false: A continuous function has an antiderivative.

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Problem 530

(a) $\lim _{x \rightarrow 0} \frac{\sin x}{\sin x x}$

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Problem 531

חשבו את הגבול הבא: $\lim _{n \rightarrow \infty}\left(1+\frac{13}{n^{2}}\right)^{\frac{(9 \cdot n+1)^{2}}{2}}=$

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Problem 532

Question 3 Not yet answered
The value of $a$ such that $f(x) = \begin{cases} \frac{\sin(x)}{x+a}, & x < 0 \\ x^2 + 2x, & x \ge 0 \end{cases}$ has a jump discontinuity at $x = 0$ is 1 0 3 2 ( 101 MATH) تجميع تفاضل وتكامل (1) Time left 0:43:23

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Problem 533

Given the Beverton-Holt model $R(N_t)=\frac{R_0}{1+a N_t}$ with $a=0.05$ , $R_0=5$ , find $N_t$ for $t=1,2,\ldots,5$ and $\lim_{t \to \infty} N_t$ for $N_0=2$ .

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Problem 534

Use the Beverton-Holt model $R(N_{t})=\frac{R_{0}}{1+a N_{t}}$ with $a=0.05$ , $R_{0}=5$ . Find $N_{t}$ for $t=1,2,\ldots,5$ and $\lim_{t \to \infty} N_{t}$ for $N_{0}=2$ .

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Problem 535

$f(x) = \begin{cases} -1 + \frac{\sqrt{x^2+1}-1}{x} & \text{si } x < 0 \\ 1 + \frac{x^2+3x}{x+1} & \text{si } x > 0 \\ 0 & \text{si } x = 0 \end{cases}$
Soit la fonction $f$ définie par : ... $f(0) = 0$
1. a) Calculer le limites de $f$ en $-\infty$ . Interpréter graphiquement le résultat obtenu.
b) Calculer les limites de $f$ en $+\infty$ . Montrer que la droite dont une équation : $y = x+3$ est une asymptote à ...
(C) au voisinage de $+\infty$ .
2. Etudier la continuité de $f$ en $0$ .

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Problem 536

1. Evaluate $\operatorname{Lim}_{x \rightarrow 2}\left(\frac{x^{2}-x-2}{x^{2}-4}\right)$
2. Evaluate $\operatorname{Lim}_{x \rightarrow 3}\left(\frac{x^{2}-1}{x-1}\right)$
3. Find the limits of the polynomial $\operatorname{Lim}_{x \rightarrow 2}\left(\frac{x^{2}+x-2}{x^{2}-3 x+2}\right)$
4. Evaluate $\operatorname{Lim}_{1} \cdots\left(\frac{x^{3}-2 x+1}{3 x^{3}+4 x^{2}-1}\right)$
5 Evaluate $\operatorname{Lim}_{101}\left(\frac{\operatorname{Sin} x}{x}\right)$ (1). Using the L'Hospital Rule, solve the limit Evaluate $\operatorname{Lim}_{x \rightarrow( }\binom{x^{3}+2 x^{2}-4 x+7}{5 x^{3}-4 x^{2}+8 x-9}$
7. Limit $\operatorname{Sin} 3$ - "." $\operatorname{Sin} 4 . x$ $\therefore \operatorname{limmit} \frac{\left(13 x^{2}-5 x+1\right)-3}{x-1}$
4. $\operatorname{limit}^{x^{2}-(a+1) x+a} \begin{array}{c}x^{3}-a^{3}\end{array}$
10. $\operatorname{Limit}_{x \rightarrow 10} \frac{\log x}{x}$

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Problem 537

2) $\lim_{x \to 1} \frac{\ln{x^2}}{x^2 - 1}$

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Problem 538

The graph of $y=f(x)$ is shown to the right. Identify the intervals on which $f(x)$ is increasing.
Which of the following shows every interval on which $f(x)$ is increasing? Choose the correct answer below. A. $(b, c),(d, e),(f, h)$ B. $(b, d),(f, g)$ C. $(b, d),(f, h)$ D. $(\mathrm{b}, \mathrm{c}),(\mathrm{d}, \mathrm{e}),(\mathrm{f}, \mathrm{g})$

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Problem 539

Exer. 21-24: Use polar coordinates to find the limit, if it exists. $21 \lim _{(x, y) \rightarrow(0,0)} \frac{x y^{2}}{x^{2}+y^{2}}$ $22 \lim _{(x, y) \rightarrow(0,0)} \frac{x^{3}-y^{3}}{x^{2}+y^{2}}$ $23 \lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{\sin \left(x^{2}+y^{2}\right)}$ $24 \lim _{(4, y) \rightarrow(0,0)} \frac{\sinh \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}$

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Problem 540

Find the horizontal asymptote of the drug concentration function $C(t)=\frac{t}{7t^{2}+8}$ . What is $C=\square$ ?

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Problem 541

Find the horizontal asymptote of $C(t)=\frac{t}{7 t^{2}+8}$ , identify the graph, and determine when concentration is highest.

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Problem 542

The graph of $y = f(x)$ is shown to the right. Identify the intervals on which $f(x)$ is decreasing.
Which of the following shows every interval on which $f(x)$ is decreasing? Choose the correct answer below.
A. $(a,b)$ , $(e,f)$ , $(g,h)$ C. $(a,b)$ , $(d,e)$ , $(f,h)$ B. $(a,b)$ , $(c,e)$ , $(g,h)$ D. $(a,b)$ , $(d,f)$ , $(g,h)$

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Problem 543

1. Найти предел функции: a) $\lim _{x \rightarrow 5} \frac{x+1}{x-5}$ b) $\lim _{x \rightarrow \infty} \frac{2-x+3 x^{3}}{-4 x^{2}+9 x^{6}}$ $v_{c} \lim _{x \rightarrow 0} \frac{\sqrt{1+3 x}-\sqrt{1-2 x}}{x^{2}+5 x}$ d) $\lim _{x \rightarrow 1} \frac{x^{4}-1}{2 x^{4}-x^{2}-1}$

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Problem 544

State the open intervals over which the function is (a) increasing, (b) decreasing, and (c) constant. (a) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function is increasing over the open interval(s) $(-3,1),(1,4)$ , . $\square$ (Type your answer in interval notation. Use a comma to separate answers as needed.) B. The function is never increasing. (b) Select the correct choice below and, if necessary, fill in the answer box to complete your choice A. The function is decreasing over the open interval(s) $(-10,3)$ . $\square$ (Type your ariswer in interval notation. Use a comma to separate answers as needed.) B. The function is never decreasing. (c) Select the correct choice below and, if necessary, fill in the answer box to complete your choice A. The function is constant over the open interval(s) $\square$ (Type your answer in interval notation. Use a comma to separate answers as needed.) B. The function is never constant.

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Problem 545

It is known that $\lim_{x \to 0} \frac{\sin(2x)}{2x} = 1$ . What is $\lim_{x \to 0} \frac{\cos(5x)}{8x \cot(2x)}$ ?

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Problem 546

Determine the following limit. $\lim_{w \to \infty} \frac{10w^2 + 7w + 3}{\sqrt{25w^4 + 5w^3}}$
Select the correct choice, and, if necessary, fill in the answer box to complete your choice. A. \lim_{w \to \infty} \frac{10w^2 + 7w + 3}{\sqrt{25w^4 + 5w^3}} = \text{________} (Simplify your answer.) B. The limit does not exist and is neither $\infty$ nor $-\infty$ .

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Problem 547

Find the limit as $x$ approaches 3 for the expression $5x^{2} + 4x + 2$ .

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Problem 548

Evaluate the limit: $\lim _{x \rightarrow 196} \frac{\sqrt{x}-14}{x-196}$ to three decimal places or state DNE.

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Problem 549

Evaluate the limit or state if it doesn't exist: $\lim _{x \rightarrow 0} \frac{\sin (15 x)}{x}=$ (3 decimal places or DNE)

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Problem 550

Find the limit: $\lim_{x \rightarrow -6} \frac{x+6}{x^{2}+x-30}$ and give your answer to three decimal places.

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Problem 551

Find the one-sided limit: $\lim _{x \rightarrow 0^{-}} \frac{2 \sin (x)}{3|x|}$ to three decimal places.

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Problem 552

Find the one-sided limit: $\lim _{x \rightarrow 7^{+}} \frac{x+9}{x-7}=$ (Enter DNE if it doesn't exist.)

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Problem 553

Plot the function and estimate the limit value: $\lim _{x \rightarrow 0} \frac{\sin (2 x)}{\sin (4 x)}=$ (two decimal places).

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Problem 554

Find the left and right limits of $\frac{7 x}{x-2}$ as $x$ approaches 2: $\lim _{x \rightarrow 2^{+}}$ and $\lim _{x \rightarrow 2^{-}}$ .

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Problem 555

Find the limit: $\lim _{x \rightarrow 8^{+}} \frac{1}{x-8}$ .

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Problem 556

Find the limit: $\lim _{x \rightarrow 2^{+}} \frac{x-6}{(x-2)^{2}}$ .

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Problem 557

Find the limits for the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2}+16, & x<-16 \\ \sqrt{x+16}, & x \geq-16\end{array}\right.$ : a. $\lim _{x \rightarrow-16^{-}} f(x)$ b. $\lim _{x \rightarrow-16^{+}} f(x)$ c. $\lim _{x \rightarrow-16} f(x)$

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Problem 558

Find the limits of the function $g(x)$ defined as:
$g(x)=\begin{cases} 0 & \text{if } x \leq -6 \\ \sqrt{36-x^{2}} & \text{if } -6 < x < 6 \\ x & \text{if } x \geq 6 \end{cases}$
for $x \to -6^{-}, -6^{+}, 6^{-}, 6^{+}, 6$ .

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Problem 559

Find the limit: $\lim _{t \rightarrow 4^{+}} \frac{|7 t-28|}{t^{2}-16}$ .

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Problem 560

Find the value of $a$ so that the function $f(x)=\frac{30+x-x^{2}}{x-6}$ is continuous at $x=6$ .

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Problem 561

Check if the function $f(x)=\left\{\begin{array}{ll} \frac{x^{2}-1}{x-1} & \text{if } x \neq 1 \\ 4 & \text{if } x=1 \end{array}\right.$ is continuous at $a=1$ .

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Problem 562

Prove there’s a solution to $\cos (x)=7-2 x$ using the Intermediate Value Theorem. Find intervals for $a$ and $b$ .

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Problem 563

Find the limit: $\lim _{x \rightarrow 4^{+}} \frac{3-x}{x^{2}-2 x-8}$ .

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Problem 564

Find the average velocity of a ball rolling down a ramp for functions $s(t)=10 t^{2}$ and $s(t)=12 t^{2}$ over specified intervals.

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Problem 565

$\lim _{x \rightarrow 0} \frac{\ln (1+x)+x}{x^{2}}$

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Problem 566

The function $f(x)$ is twice differentiable on the closed interval $[-3, 4]$ . Selected values of $f(x)$ , $f'(x)$ , and $f''(x)$ are given in the table above.
A) Show that there must be a value $c$ , $-3 < c < 4$ , such that $f(c) = 1$ ? Justify your answer.

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Problem 567

$\lim _{x \rightarrow-1} \frac{x^{3}+4 x^{2}-3}{x^{5}+5}$

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Problem 568

1) lim x

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Problem 569

Question Use the graph of $h(x)$ shown below to evaluate $\lim_{x \to -3} h(x)$ , if possible. If the limit does not exist, enter Ø.

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Problem 570

Question What is $a$ , $f(x)$ , and $L$ given $f(x)$ approaches $L$ as $x$ approaches $a$ in the following limit?
$\lim_{x \to 3} (x^2 - 8x - 3) = -18$
Provide your answer below:
$a =$ $f(x) =$ $L =$
FEEDBACK MORE INSTRUCTION

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Problem 571

a) $\lim_{x \to 2^-} \frac{|x^2 - x - 2|}{|x^2 - 5x + 6|}$

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Problem 572

$\lim _{x \rightarrow 0} \frac{e^{3 x}+e^{5 x}+e^{7 x}-3}{9 x}$

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Problem 573

أحسب نهايات الدالة g

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Problem 574

Question Find the two one-sided limits as $x \to 1$ of the ceil function
$f(x) = \lceil x \rceil = \begin{cases} -1 & \text{if } -2 < x \le -1 \\ 0 & \text{if } -1 < x \le 0 \\ 1 & \text{if } 0 < x \le 1 \\ 2 & \text{if } 1 < x \le 2 \\ 3 & \text{if } 2 < x \le 3 \\ \vdots \end{cases}$
If a limit does not exist, enter $\emptyset$ as your answer.
Provide your answer below:
a. $\lim_{x \to 1^-} f(x) =$ b. $\lim_{x \to 1^+} f(x) =$

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Problem 575

Question Evaluate $\lim_{x \to 3} \frac{-3x^3 - 2}{-3x - 3}$ Enter an exact answer. Provide your answer below:

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Problem 576

1. $\lim _{x \rightarrow-1} \frac{x+1}{\sqrt{x+5}-2}$

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Problem 577

Question The graph of $f(x)$ is shown below. Which of the following statements are true?
Select the correct answer below: $f(x)$ has a removable discontinuity at $x = -3$ . $f(x)$ has a jump discontinuity at $x = -3$ . $f(x)$ has an infinite discontinuity at $x = -3$ . $f(x)$ is continuous at $x = -3$ .

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Problem 578

Question The graph of $f(x)$ is shown below. Which of the following statements are true?
Select the correct answer below: $f(x)$ has a removable discontinuity at $x = 0$ . $f(x)$ has a jump discontinuity at $x = 0$ . $f(x)$ has an infinite discontinuity at $x = 0$ . $f(x)$ is continuous at $x = 0$ . FEEDBACK MORE INSTRUCTION SUBMIT

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Problem 579

Question Given $f(x) = \frac{-2x^2 - 6x + 20}{x - 4}$ , which of the following statements are true?
Select the correct answer below: $f(x)$ has a removable discontinuity at $x = 4$ . $f(x)$ has a jump discontinuity at $x = 4$ $f(x)$ has an infinite discontinuity at $x = 4$ . $f(x)$ is continuous at $x = 4$

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Problem 580

Find the limit as $x$ approaches 4 for the expression $x^{3}+x$ .

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Problem 581

Evaluate these limits: 1. $\lim_{x \to 4}(x^3 + x)$ 2. $\lim_{x \to 4}(x^2 + 1)$ 3. $\lim_{x \to 2} \frac{x^2 - x - 2}{x^2 - 2x}$ 4. $\lim_{x \to 1} \frac{x^2 - 2x + 1}{x^3 - x}$ 5. $\lim_{x \to \infty} \frac{\sqrt{x^2 + 1}}{x^2}$ 6. $\lim_{x \to 4}(x^2 + 3x - 5)$ 7. $\lim_{y \to \infty}(y^3 - 2y + 7)$ 8. $\lim_{t \to 0} \frac{2t^2 + 1}{t^3 + 3t - 4}$ 9. $\lim_{x \to 1} \frac{(s + 1)^2}{2x^2 + 3}$ 10. $\lim_{w \to 2} \frac{3w^2 - 4w + 2}{w^3 - 5}$ 11. $\lim_{w \to -1} \frac{3w^2 - 2w + 7}{w^2 + 1}$ 12. $\lim_{x \to 2} \frac{\sqrt{x - 2}}{\sqrt{x^2 - 4}}$ 13. $\lim_{x \to 2} \frac{(1 - x^2)^{1}}{(1 - x^2)^2}$ 14. $\lim_{x \to 3} \frac{x - 3}{\sqrt{x^2 - 9}}$ 15. $\lim_{x \to \infty} \frac{4}{2x^2 - 3}$ 16. $\lim_{x \to \infty} \frac{2x^2}{3x^2 + 5}$ 17. $\lim_{x \to \infty} \frac{x^2}{3x^2 - 4x + 1}$ 18. $\lim_{x \to \infty} 2^{\frac{3}{x}}$ 19. $\lim_{x \to 0^+} 2^{\frac{1}{x}}$ 20. $\lim_{x \to 0^+} \frac{1}{1 + 2^{\frac{1}{x}}}$

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Problem 582

(c) $\lim_{x \to 2} \frac{3x^2 - x - 10}{2x^2 + 3x - 14}$

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Problem 583

$\lim_{x \to 1} \frac{(100\pi^q - 981)(y - \sqrt{2})(e^p - e^\pi)(x^{\pi+1} - x^\pi)}{(z^z - \sqrt{2}^{\sqrt{2}})(m^m \cdot e^{1/e} - 1)(d^d \cdot \pi^{1/\pi} - 1)(x^{1+e} - x^e)}$
Task: Find the values of $p$ , $q$ , $y$ , $m$ , $d$ , and $z$ such that the above limit expression is indeterminate as $x \to 1$ .

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Problem 584

$\lim _{x \rightarrow \infty} \frac{\tan ^{-1}(x)}{(1 / x)-4}=$

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Problem 585

2. Визначить тип точок розриву функції $f(x) = \frac{1}{x} \text{arctg} \frac{1}{1-x}$ , $x \in R$ на $D_f$ .

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Problem 586

3. Дослідить функцію $f(x) = \frac{x^2 + 1}{\sqrt{x}}$ на рівномірну неперервність на множині $X = (10, +\infty)$

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Problem 587

4. Знайдіть границю: $\lim_{x\to 0} \frac{\text{ch}(xe^x) - \text{ch}(xe^{-x})}{x^3}.$

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Problem 588

Question 3, 11.2.37 21 points 0 Points: 0 of 1 Save
Use a calculator to estimate the given limit. $\lim _{x \rightarrow-\infty} \frac{x^{2 / 3}-x^{5 / 3}}{x^{3}}$
What is the limit? Select the correct choice below and, if necessary fill in the answer box to complete your choice. A. $\lim _{x \rightarrow-\infty} \frac{x^{2 / 3}-x^{5 / 3}}{x^{3}}=$ $\square$ B. The limit does not exist and is not $\infty$ or $-\infty$ .

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Problem 589

c. $\lim_{x \to 2^{-}} \frac{x+2}{x-2}$ $= \frac{2}{0^{-}}$ 2 marks $= -\infty$ 1 mark

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Problem 590

Find the following limit or state that it does not exist. $\lim _{x \rightarrow 2} 5$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $\lim _{x \rightarrow 2} 5=\square$ (Simplify your answer.) B. The limit does not exist.

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Problem 591

$\lim _{x \rightarrow 5} \frac{x^{3}-125}{x-5}$

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Problem 592

Find the limit of $f(x)$ as $x$ approaches 1, where $f(x)=\frac{(x-1)^{2}(x+1)}{|x-1|}$ for $x \neq 1$ and $f(1)=2$ .

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Problem 593

Find $\lim _{x \rightarrow \frac{\pi}{4}} g(x)$ for $g(x)=\frac{2 \cos ^{2} x-1}{\cos x-\sin x}$ .

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Problem 594

Find $\lim _{x \rightarrow \frac{\pi}{4}} g(x)$ for $g(x)=\frac{2 \cos ^{2} x-1}{\cos x-\sin x}$ . Which option is equivalent?

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Problem 595

Find $\lim_{x \rightarrow 1} f(x)$ if $g(x) \leq f(x) \leq h(x)$ where $g(x)=\sin \left(\frac{\pi}{2} x\right)+4$ and $h(x)=-\frac{1}{4} x^{3}+\frac{3}{4} x+\frac{9}{2}$ .

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Problem 596

Calculate the average rate of change of $g(x)=-5 x^{3}+4$ between $x=-4$ and $x=4$ .

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Problem 597

2. [4 marks] Given the piece-wise function $g(x)$ defined as $g(x) = \begin{cases} x^2 + 3 & \text{if } x < 3 \\ 15 - x & \text{if } x > 3 \end{cases}$ .
find $\lim_{x \to 3} g(x)$ , if it exists. Show all your work.
$\lim_{x \to 3} g(x)$ exists at $g(x) = 12$

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Problem 598

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. $\lim _{x \rightarrow 7} \frac{x}{x+2}$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $\lim _{x \rightarrow 7} \frac{x}{x+2}=$ $\square$ (Simplify your answer. Type an integer or a simplified fraction.) B. The limit does not exist.

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Problem 599

The graph below is the function $f(x)$
Find $\lim_{x \to 4^-} f(x)$
Find $\lim_{x \to 4^+} f(x)$
Find $\lim_{x \to 4} f(x)$ Enter an integer or decimal number [more..]

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Problem 600

Find the limit. Use l'Hospital's Rule where a $\lim _{x \rightarrow \infty} x^{6 / x}$

See Solution

Limits & Continuity

Problem 501

Use the graph of the function f(x)f(x)f(x) shown below in order to determine all values of aaa for which lim⁡x→af(x)=−3\lim_{x \to a} f(x) = -3limx→a​f(x)=−3 on the interval −9<x<9-9 < x < 9−9<x<9.Answer Attempt 2 out of 2Additional Solution No Solutiona=a =a=

Problem 502

Over which interval does function f(x)=1x+5f(x) = \frac{1}{\sqrt{x+5}}f(x)=x+5​1​ have a point (points) of discontinuity?(A) ]-5, +∞\infty∞ [ (B) ]-∞\infty∞, -5] (C) ]0, -5[ (D) ]0, -5[ (E) [-5, +∞\infty∞[

Problem 503

Problem 504

Estimate the soccer ball's instantaneous velocity at t=2t=2t=2 s using the average velocities given. Round to two decimal places.

Problem 505

Prove that lim⁡x→64x=24\lim _{x \rightarrow 6} 4 x=24limx→6​4x=24 using the precise definition of a limit.

Problem 506

Find the average velocity vˉ\bar{v}vˉ of a stone with height h(t)=30t−4.9t2h(t)=30t-4.9t^2h(t)=30t−4.9t2 over intervals around t=3t=3t=3.

Problem 507

Evaluate the function f(x)f(x)f(x) and determine which statements about its limit and continuity at x=1x=1x=1 are true. Options: A. Only I, B. Only II, C. I and II, D. None, E. All.

Problem 508

Problem 509

21. lim⁡x→∞4−x2+x\lim _{x \rightarrow \infty} \frac{4-\sqrt{x}}{2+\sqrt{x}}limx→∞​2+x​4−x​​23. lim⁡x→∞x+3x24x−1\lim _{x \rightarrow \infty} \frac{\sqrt{x+3 x^{2}}}{4 x-1}limx→∞​4x−1x+3x2​​

Problem 510

Problem 511

Problem 512

20.lim⁡x→π2ln⁡xtan⁡(x2)20. \lim_{x \to \frac{\pi}{2}} \ln{x} \tan{(\frac{x}{2})}20.limx→2π​​lnxtan(2x​)

Problem 513

Problem 514

Problem 515

Problem 516

lim⁡x→−34−x2+73x+9\lim _{x \rightarrow-3} \frac{4-\sqrt{x^{2}+7}}{3 x+9}limx→−3​3x+94−x2+7​​

Problem 517

Find the limits: 1. lim⁡x→33−xx2−9\lim _{x \rightarrow 3} \frac{3-x}{x^{2}-9}limx→3​x2−93−x​ and 2. lim⁡x→4x2−5x+4x2−2x−8\lim _{x \rightarrow 4} \frac{x^{2}-5 x+4}{x^{2}-2 x-8}limx→4​x2−2x−8x2−5x+4​.

Problem 518

Find the limit as x approaches 1: lim⁡x→12x3+x+1−2xx−1\lim_{{x \to 1}} \frac{\sqrt{2 x^{3}+x+1}-2 x}{x-1}limx→1​x−12x3+x+1​−2x​

Problem 519

Problem 520

Find the limit as hhh approaches 0 for the difference quotient of: 1. f(x)=x2−xf(x)=x^{2}-xf(x)=x2−x, 2. f(x)=1x+3f(x)=\frac{1}{x+3}f(x)=x+31​.

Problem 521

מה הגבול של (x3+x23−x)\left(\sqrt[3]{x^{3}+x^{2}}-x\right)(3x3+x2​−x) כש-xxx מתקרב למינוס אינסוף?

Problem 522

Evaluate the piecewise function fff at x=3x=3x=3 to check if it's continuous and differentiable. What is true about fff at x=3x=3x=3?

Problem 523

Problem 524

Given g(x)g(x)g(x) is differentiable at x=−3x=-3x=−3 with g(−3)=9g(-3)=9g(−3)=9 and g′(−3)=0g'(-3)=0g′(−3)=0, which statements are true?

Problem 525

Given the piecewise function fff, determine if it is continuous or differentiable at x=−3x=-3x=−3. Options: 1) neither, 2) continuous only, 3) both.

Problem 526

Is the function f(x)={sin⁡(x)x<π22xπx≥π2f(x)=\left\{\begin{array}{cc}\sin (x) & x<\frac{\pi}{2} \\ \frac{2 x}{\pi} & x \geq \frac{\pi}{2}\end{array}\right.f(x)={sin(x)π2x​​x<2π​x≥2π​​ continuous at x=π2x=\frac{\pi}{2}x=2π​? Explain.

Problem 527

Find the acceleration due to gravity at 6989 m, the horizontal asymptote of g(h)g(h)g(h), and solve g(h)=0g(h)=0g(h)=0.

Problem 528

When can average speed equal instantaneous speed?

Problem 529

True or false: A continuous function has an antiderivative.

Problem 530

(a) lim⁡x→0sin⁡xsin⁡xx\lim _{x \rightarrow 0} \frac{\sin x}{\sin x x}limx→0​sinxxsinx​

Problem 531

חשבו את הגבול הבא: lim⁡n→∞(1+13n2)(9⋅n+1)22=\lim _{n \rightarrow \infty}\left(1+\frac{13}{n^{2}}\right)^{\frac{(9 \cdot n+1)^{2}}{2}}=n→∞lim​(1+n213​)2(9⋅n+1)2​=

Problem 532

Problem 533

Given the Beverton-Holt model R(Nt)=R01+aNtR(N_t)=\frac{R_0}{1+a N_t}R(Nt​)=1+aNt​R0​​ with a=0.05a=0.05a=0.05, R0=5R_0=5R0​=5, find NtN_tNt​ for t=1,2,…,5t=1,2,\ldots,5t=1,2,…,5 and lim⁡t→∞Nt\lim_{t \to \infty} N_tlimt→∞​Nt​ for N0=2N_0=2N0​=2.

Problem 534

Use the Beverton-Holt model R(Nt)=R01+aNtR(N_{t})=\frac{R_{0}}{1+a N_{t}}R(Nt​)=1+aNt​R0​​ with a=0.05a=0.05a=0.05, R0=5R_{0}=5R0​=5. Find NtN_{t}Nt​ for t=1,2,…,5t=1,2,\ldots,5t=1,2,…,5 and lim⁡t→∞Nt\lim_{t \to \infty} N_{t}limt→∞​Nt​ for N0=2N_{0}=2N0​=2.

Problem 535

Problem 536

Problem 537

2) lim⁡x→1ln⁡x2x2−1\lim_{x \to 1} \frac{\ln{x^2}}{x^2 - 1}limx→1​x2−1lnx2​

Problem 538

Problem 539

Problem 540

Find the horizontal asymptote of the drug concentration function C(t)=t7t2+8C(t)=\frac{t}{7t^{2}+8}C(t)=7t2+8t​. What is C=□C=\squareC=□?

Problem 541

Find the horizontal asymptote of C(t)=t7t2+8C(t)=\frac{t}{7 t^{2}+8}C(t)=7t2+8t​, identify the graph, and determine when concentration is highest.

Problem 542

Problem 543

Problem 544

Problem 545

It is known that lim⁡x→0sin⁡(2x)2x=1\lim_{x \to 0} \frac{\sin(2x)}{2x} = 1limx→0​2xsin(2x)​=1. What is lim⁡x→0cos⁡(5x)8xcot⁡(2x)\lim_{x \to 0} \frac{\cos(5x)}{8x \cot(2x)}limx→0​8xcot(2x)cos(5x)​?

Problem 546

Problem 547

Find the limit as xxx approaches 3 for the expression 5x2+4x+25x^{2} + 4x + 25x2+4x+2.

Problem 548

Evaluate the limit: lim⁡x→196x−14x−196\lim _{x \rightarrow 196} \frac{\sqrt{x}-14}{x-196}x→196lim​x−196x​−14​ to three decimal places or state DNE.

Problem 549

Use the graph of the function $f(x)$ shown below in order to determine all values of $a$ for which $\lim_{x \to a} f(x) = -3$ on the interval $-9 < x < 9$ .
Answer Attempt 2 out of 2
Additional Solution No Solution
$a =$

Over which interval does function $f(x) = \frac{1}{\sqrt{x+5}}$ have a point (points) of discontinuity?
(A) ]-5, + $\infty$ [ (B) ]- $\infty$ , -5] (C) ]0, -5[ (D) ]0, -5[ (E) [-5, + $\infty$ [

Estimate the soccer ball's instantaneous velocity at $t=2$ s using the average velocities given. Round to two decimal places.

Prove that $\lim _{x \rightarrow 6} 4 x=24$ using the precise definition of a limit.

Find the average velocity $\bar{v}$ of a stone with height $h(t)=30t-4.9t^2$ over intervals around $t=3$ .

Evaluate the function $f(x)$ and determine which statements about its limit and continuity at $x=1$ are true. Options: A. Only I, B. Only II, C. I and II, D. None, E. All.

21. $\lim _{x \rightarrow \infty} \frac{4-\sqrt{x}}{2+\sqrt{x}}$
23. $\lim _{x \rightarrow \infty} \frac{\sqrt{x+3 x^{2}}}{4 x-1}$

$20. \lim_{x \to \frac{\pi}{2}} \ln{x} \tan{(\frac{x}{2})}$

$\lim _{x \rightarrow-3} \frac{4-\sqrt{x^{2}+7}}{3 x+9}$

Find the limits: 1. $\lim _{x \rightarrow 3} \frac{3-x}{x^{2}-9}$ and 2. $\lim _{x \rightarrow 4} \frac{x^{2}-5 x+4}{x^{2}-2 x-8}$ .

Find the limit as x approaches 1: $\lim_{{x \to 1}} \frac{\sqrt{2 x^{3}+x+1}-2 x}{x-1}$

Find the limit as $h$ approaches 0 for the difference quotient of: 1. $f(x)=x^{2}-x$ , 2. $f(x)=\frac{1}{x+3}$ .

מה הגבול של $\left(\sqrt[3]{x^{3}+x^{2}}-x\right)$ כש- $x$ מתקרב למינוס אינסוף?

Evaluate the piecewise function $f$ at $x=3$ to check if it's continuous and differentiable. What is true about $f$ at $x=3$ ?

Given $g(x)$ is differentiable at $x=-3$ with $g(-3)=9$ and $g'(-3)=0$ , which statements are true?

Given the piecewise function $f$ , determine if it is continuous or differentiable at $x=-3$ . Options: 1) neither, 2) continuous only, 3) both.

Is the function $f(x)=\left\{\begin{array}{cc}\sin (x) & x<\frac{\pi}{2} \\ \frac{2 x}{\pi} & x \geq \frac{\pi}{2}\end{array}\right.$ continuous at $x=\frac{\pi}{2}$ ? Explain.

Find the acceleration due to gravity at 6989 m, the horizontal asymptote of $g(h)$ , and solve $g(h)=0$ .

(a) $\lim _{x \rightarrow 0} \frac{\sin x}{\sin x x}$

חשבו את הגבול הבא: $\lim _{n \rightarrow \infty}\left(1+\frac{13}{n^{2}}\right)^{\frac{(9 \cdot n+1)^{2}}{2}}=$

Given the Beverton-Holt model $R(N_t)=\frac{R_0}{1+a N_t}$ with $a=0.05$ , $R_0=5$ , find $N_t$ for $t=1,2,\ldots,5$ and $\lim_{t \to \infty} N_t$ for $N_0=2$ .

Use the Beverton-Holt model $R(N_{t})=\frac{R_{0}}{1+a N_{t}}$ with $a=0.05$ , $R_{0}=5$ . Find $N_{t}$ for $t=1,2,\ldots,5$ and $\lim_{t \to \infty} N_{t}$ for $N_{0}=2$ .

2) $\lim_{x \to 1} \frac{\ln{x^2}}{x^2 - 1}$

Find the horizontal asymptote of the drug concentration function $C(t)=\frac{t}{7t^{2}+8}$ . What is $C=\square$ ?

Find the horizontal asymptote of $C(t)=\frac{t}{7 t^{2}+8}$ , identify the graph, and determine when concentration is highest.

It is known that $\lim_{x \to 0} \frac{\sin(2x)}{2x} = 1$ . What is $\lim_{x \to 0} \frac{\cos(5x)}{8x \cot(2x)}$ ?

Find the limit as $x$ approaches 3 for the expression $5x^{2} + 4x + 2$ .

Evaluate the limit: $\lim _{x \rightarrow 196} \frac{\sqrt{x}-14}{x-196}$ to three decimal places or state DNE.

Evaluate the limit or state if it doesn't exist: $\lim _{x \rightarrow 0} \frac{\sin (15 x)}{x}=$ (3 decimal places or DNE)

Find the limit: $\lim_{x \rightarrow -6} \frac{x+6}{x^{2}+x-30}$ and give your answer to three decimal places.

Find the one-sided limit: $\lim _{x \rightarrow 0^{-}} \frac{2 \sin (x)}{3|x|}$ to three decimal places.

Find the one-sided limit: $\lim _{x \rightarrow 7^{+}} \frac{x+9}{x-7}=$ (Enter DNE if it doesn't exist.)

Plot the function and estimate the limit value: $\lim _{x \rightarrow 0} \frac{\sin (2 x)}{\sin (4 x)}=$ (two decimal places).

Find the left and right limits of $\frac{7 x}{x-2}$ as $x$ approaches 2: $\lim _{x \rightarrow 2^{+}}$ and $\lim _{x \rightarrow 2^{-}}$ .

Find the limit: $\lim _{x \rightarrow 8^{+}} \frac{1}{x-8}$ .

Find the limit: $\lim _{x \rightarrow 2^{+}} \frac{x-6}{(x-2)^{2}}$ .

Find the limit: $\lim _{t \rightarrow 4^{+}} \frac{|7 t-28|}{t^{2}-16}$ .

Find the value of $a$ so that the function $f(x)=\frac{30+x-x^{2}}{x-6}$ is continuous at $x=6$ .

Check if the function $f(x)=\left\{\begin{array}{ll} \frac{x^{2}-1}{x-1} & \text{if } x \neq 1 \\ 4 & \text{if } x=1 \end{array}\right.$ is continuous at $a=1$ .

Prove there’s a solution to $\cos (x)=7-2 x$ using the Intermediate Value Theorem. Find intervals for $a$ and $b$ .

Find the limit: $\lim _{x \rightarrow 4^{+}} \frac{3-x}{x^{2}-2 x-8}$ .

Find the average velocity of a ball rolling down a ramp for functions $s(t)=10 t^{2}$ and $s(t)=12 t^{2}$ over specified intervals.

$\lim _{x \rightarrow 0} \frac{\ln (1+x)+x}{x^{2}}$

$\lim _{x \rightarrow-1} \frac{x^{3}+4 x^{2}-3}{x^{5}+5}$

Question Use the graph of $h(x)$ shown below to evaluate $\lim_{x \to -3} h(x)$ , if possible. If the limit does not exist, enter Ø.

a) $\lim_{x \to 2^-} \frac{|x^2 - x - 2|}{|x^2 - 5x + 6|}$

$\lim _{x \rightarrow 0} \frac{e^{3 x}+e^{5 x}+e^{7 x}-3}{9 x}$