QuestionFind the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit of the function $\frac{\sqrt{4+x}-}{x}$ as $x$ approaches $0$ . . The function is defined for all real numbers except $x =0$ .

STEP 2

We can see that if we substitute $x =0$ directly into the function, we will get an indeterminate form of type $\frac{0}{0}$ . To resolve this, we can use the conjugate of the numerator to simplify the function.
The conjugate of $\sqrt{4+x}-2$ is $\sqrt{4+x}+2$ .

STEP 3

Multiply the numerator and the denominator by the conjugate of the numerator.
$\lim{x \rightarrow0} \frac{\sqrt{+x}-2}{x} \times \frac{\sqrt{+x}+2}{\sqrt{+x}+2}$

STEP 4

implify the numerator using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ .
$\lim{x \rightarrow0} \frac{(\sqrt{4+x})^2 -2^2}{x(\sqrt{4+x}+2)}$

STEP 5

implify the expression further.
$\lim{x \rightarrow0} \frac{4+x -4}{x(\sqrt{4+x}+2)}$

STEP 6

implify the numerator.
$\lim{x \rightarrow0} \frac{x}{x(\sqrt{4+x}+2)}$

STEP 7

Cancel out the common factor $x$ from the numerator and the denominator.
$\lim{x \rightarrow0} \frac{1}{\sqrt{4+x}+2}$

STEP 8

Now, substitute $x =0$ into the simplified function.
$\lim{x \rightarrow0} \frac{1}{\sqrt{4+0}+2}$

STEP 9

implify the expression to get the limit.
$\lim{x \rightarrow} \frac{}{\sqrt{4+}+2} = \frac{}{2+2} = \frac{}{4}$ So, $\lim{x \rightarrow} \frac{\sqrt{4+x}-2}{x} = \frac{}{4}$ .

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QuestionFind the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit of the function $\frac{\sqrt{4+x}-}{x}$ as $x$ approaches $0$ . . The function is defined for all real numbers except $x =0$ .

STEP 2

We can see that if we substitute $x =0$ directly into the function, we will get an indeterminate form of type $\frac{0}{0}$ . To resolve this, we can use the conjugate of the numerator to simplify the function.
The conjugate of $\sqrt{4+x}-2$ is $\sqrt{4+x}+2$ .

STEP 3

Multiply the numerator and the denominator by the conjugate of the numerator.
$\lim{x \rightarrow0} \frac{\sqrt{+x}-2}{x} \times \frac{\sqrt{+x}+2}{\sqrt{+x}+2}$

STEP 4

implify the numerator using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ .
$\lim{x \rightarrow0} \frac{(\sqrt{4+x})^2 -2^2}{x(\sqrt{4+x}+2)}$

STEP 5

implify the expression further.
$\lim{x \rightarrow0} \frac{4+x -4}{x(\sqrt{4+x}+2)}$

STEP 6

implify the numerator.
$\lim{x \rightarrow0} \frac{x}{x(\sqrt{4+x}+2)}$

STEP 7

Cancel out the common factor $x$ from the numerator and the denominator.
$\lim{x \rightarrow0} \frac{1}{\sqrt{4+x}+2}$

STEP 8

Now, substitute $x =0$ into the simplified function.
$\lim{x \rightarrow0} \frac{1}{\sqrt{4+0}+2}$

STEP 9

implify the expression to get the limit.
$\lim{x \rightarrow} \frac{}{\sqrt{4+}+2} = \frac{}{2+2} = \frac{}{4}$ So, $\lim{x \rightarrow} \frac{\sqrt{4+x}-2}{x} = \frac{}{4}$ .

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QuestionFind the limit: lim⁡x→04+x−2x\lim _{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x}limx→0​x4+x​−2​.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit of the function 4+x−x\frac{\sqrt{4+x}-}{x}x4+x​−​ as xxx approaches 000. . The function is defined for all real numbers except x=0x =0x=0.

STEP 2

We can see that if we substitute x=0x =0x=0 directly into the function, we will get an indeterminate form of type 00\frac{0}{0}00​. To resolve this, we can use the conjugate of the numerator to simplify the function.The conjugate of 4+x−2\sqrt{4+x}-24+x​−2 is 4+x+2\sqrt{4+x}+24+x​+2.

STEP 3

Multiply the numerator and the denominator by the conjugate of the numerator.lim⁡x→0+x−2x×+x+2+x+2\lim{x \rightarrow0} \frac{\sqrt{+x}-2}{x} \times \frac{\sqrt{+x}+2}{\sqrt{+x}+2}limx→0x+x​−2​×+x​+2+x​+2​

STEP 4

implify the numerator using the difference of squares formula (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2(a−b)(a+b)=a2−b2.lim⁡x→0(4+x)2−22x(4+x+2)\lim{x \rightarrow0} \frac{(\sqrt{4+x})^2 -2^2}{x(\sqrt{4+x}+2)}limx→0x(4+x​+2)(4+x​)2−22​

STEP 5

implify the expression further.lim⁡x→04+x−4x(4+x+2)\lim{x \rightarrow0} \frac{4+x -4}{x(\sqrt{4+x}+2)}limx→0x(4+x​+2)4+x−4​

STEP 6

implify the numerator.lim⁡x→0xx(4+x+2)\lim{x \rightarrow0} \frac{x}{x(\sqrt{4+x}+2)}limx→0x(4+x​+2)x​

STEP 7

Cancel out the common factor xxx from the numerator and the denominator.lim⁡x→014+x+2\lim{x \rightarrow0} \frac{1}{\sqrt{4+x}+2}limx→04+x​+21​

STEP 8

Now, substitute x=0x =0x=0 into the simplified function.lim⁡x→014+0+2\lim{x \rightarrow0} \frac{1}{\sqrt{4+0}+2}limx→04+0​+21​

STEP 9

implify the expression to get the limit.lim⁡x→4++2=2+2=4\lim{x \rightarrow} \frac{}{\sqrt{4+}+2} = \frac{}{2+2} = \frac{}{4}limx→4+​+2​=2+2​=4​So, lim⁡x→4+x−2x=4\lim{x \rightarrow} \frac{\sqrt{4+x}-2}{x} = \frac{}{4}limx→x4+x​−2​=4​.

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QuestionFind the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x}$ .

Assumptions1. We are asked to find the limit of the function $\frac{\sqrt{4+x}-}{x}$ as $x$ approaches $0$ . . The function is defined for all real numbers except $x =0$ .

We can see that if we substitute $x =0$ directly into the function, we will get an indeterminate form of type $\frac{0}{0}$ . To resolve this, we can use the conjugate of the numerator to simplify the function.
The conjugate of $\sqrt{4+x}-2$ is $\sqrt{4+x}+2$ .

Multiply the numerator and the denominator by the conjugate of the numerator.
$\lim{x \rightarrow0} \frac{\sqrt{+x}-2}{x} \times \frac{\sqrt{+x}+2}{\sqrt{+x}+2}$

implify the numerator using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ .
$\lim{x \rightarrow0} \frac{(\sqrt{4+x})^2 -2^2}{x(\sqrt{4+x}+2)}$

implify the expression further.
$\lim{x \rightarrow0} \frac{4+x -4}{x(\sqrt{4+x}+2)}$

implify the numerator.
$\lim{x \rightarrow0} \frac{x}{x(\sqrt{4+x}+2)}$

Cancel out the common factor $x$ from the numerator and the denominator.
$\lim{x \rightarrow0} \frac{1}{\sqrt{4+x}+2}$

Now, substitute $x =0$ into the simplified function.
$\lim{x \rightarrow0} \frac{1}{\sqrt{4+0}+2}$

implify the expression to get the limit.
$\lim{x \rightarrow} \frac{}{\sqrt{4+}+2} = \frac{}{2+2} = \frac{}{4}$ So, $\lim{x \rightarrow} \frac{\sqrt{4+x}-2}{x} = \frac{}{4}$ .