Math / Calculus

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

Studdy Solution

STEP 1

What is this asking? We need to find what the expression approaches as $x$ gets really close to 0. Watch out! If we just plug in $x = 0$ , we get $\frac{0}{0}$ , which is undefined!
So, we need a clever trick.

STEP 2

1. Rewrite the expression
2. Evaluate the limit

STEP 3

Let's multiply the expression by $1$ in a sneaky way.
We'll use the conjugate of the numerator: $\sqrt{4x + 49} + 7$ .
This will help us simplify the expression!

STEP 4

$\begin{aligned} \lim_{x \rightarrow 0} \frac{\sqrt{4x + 49} - 7}{x} &= \lim_{x \rightarrow 0} \frac{(4x + 49) - 49}{x(\sqrt{4x + 49} + 7)} \\ &= \lim_{x \rightarrow 0} \frac{4x}{x(\sqrt{4x + 49} + 7)} \end{aligned}$

STEP 5

Now, we can divide both the numerator and denominator by $x$ (since $x$ is approaching 0, it's not equal to 0, so this is valid). $\lim_{x \rightarrow 0} \frac{4x}{x(\sqrt{4x + 49} + 7)} = \lim_{x \rightarrow 0} \frac{4}{\sqrt{4x + 49} + 7}$ Look at that, the pesky $x$ in the denominator is gone!

STEP 6

Now we can substitute $x = 0$ into the simplified expression: $\lim_{x \rightarrow 0} \frac{4}{\sqrt{4x + 49} + 7} = \frac{4}{\sqrt{4 \cdot 0 + 49} + 7}$

STEP 7

$\frac{4}{\sqrt{49} + 7} = \frac{4}{7 + 7} = \frac{4}{14} = \frac{2}{7}$ Boom!

STEP 8

The limit is $\frac{2}{7}$ .

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

Studdy Solution

STEP 1

What is this asking? We need to find what the expression approaches as $x$ gets really close to 0. Watch out! If we just plug in $x = 0$ , we get $\frac{0}{0}$ , which is undefined!
So, we need a clever trick.

STEP 2

1. Rewrite the expression
2. Evaluate the limit

STEP 3

Let's multiply the expression by $1$ in a sneaky way.
We'll use the conjugate of the numerator: $\sqrt{4x + 49} + 7$ .
This will help us simplify the expression!

STEP 4

$\begin{aligned} \lim_{x \rightarrow 0} \frac{\sqrt{4x + 49} - 7}{x} &= \lim_{x \rightarrow 0} \frac{(4x + 49) - 49}{x(\sqrt{4x + 49} + 7)} \\ &= \lim_{x \rightarrow 0} \frac{4x}{x(\sqrt{4x + 49} + 7)} \end{aligned}$

STEP 5

Now, we can divide both the numerator and denominator by $x$ (since $x$ is approaching 0, it's not equal to 0, so this is valid). $\lim_{x \rightarrow 0} \frac{4x}{x(\sqrt{4x + 49} + 7)} = \lim_{x \rightarrow 0} \frac{4}{\sqrt{4x + 49} + 7}$ Look at that, the pesky $x$ in the denominator is gone!

STEP 6

Now we can substitute $x = 0$ into the simplified expression: $\lim_{x \rightarrow 0} \frac{4}{\sqrt{4x + 49} + 7} = \frac{4}{\sqrt{4 \cdot 0 + 49} + 7}$

STEP 7

$\frac{4}{\sqrt{49} + 7} = \frac{4}{7 + 7} = \frac{4}{14} = \frac{2}{7}$ Boom!

STEP 8

The limit is $\frac{2}{7}$ .

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

Studdy Solution

STEP 1

What is this asking? We need to find what the expression approaches as xxx gets really close to 0. Watch out! If we just plug in x=0x = 0x=0, we get 00\frac{0}{0}00​, which is undefined!So, we need a clever trick.

STEP 2

1. Rewrite the expression2. Evaluate the limit

STEP 3

Let's multiply the expression by 111 in a sneaky way.We'll use the conjugate of the numerator: 4x+49+7\sqrt{4x + 49} + 74x+49​+7.This will help us simplify the expression!

STEP 4

STEP 5

STEP 6

Now we can substitute x=0x = 0x=0 into the simplified expression: lim⁡x→044x+49+7=44⋅0+49+7\lim_{x \rightarrow 0} \frac{4}{\sqrt{4x + 49} + 7} = \frac{4}{\sqrt{4 \cdot 0 + 49} + 7}limx→0​4x+49​+74​=4⋅0+49​+74​

STEP 7

449+7=47+7=414=27\frac{4}{\sqrt{49} + 7} = \frac{4}{7 + 7} = \frac{4}{14} = \frac{2}{7}49​+74​=7+74​=144​=72​ Boom!

STEP 8

The limit is 27\frac{2}{7}72​.

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Start learning now

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

What is this asking? We need to find what the expression approaches as $x$ gets really close to 0. Watch out! If we just plug in $x = 0$ , we get $\frac{0}{0}$ , which is undefined!
So, we need a clever trick.

1. Rewrite the expression
2. Evaluate the limit

Let's multiply the expression by $1$ in a sneaky way.
We'll use the conjugate of the numerator: $\sqrt{4x + 49} + 7$ .
This will help us simplify the expression!

Now we can substitute $x = 0$ into the simplified expression: $\lim_{x \rightarrow 0} \frac{4}{\sqrt{4x + 49} + 7} = \frac{4}{\sqrt{4 \cdot 0 + 49} + 7}$

$\frac{4}{\sqrt{49} + 7} = \frac{4}{7 + 7} = \frac{4}{14} = \frac{2}{7}$ Boom!

The limit is $\frac{2}{7}$ .