Math / Calculus

Question4. Evaluale mathematically $\lim _{x \rightarrow 0} \frac{x}{\sqrt{x+4-2}}$

Studdy Solution

STEP 1

1. We are tasked with evaluating the limit $\lim_{x \rightarrow 0} \frac{x}{\sqrt{x+4} - 2}$ .
2. The limit is evaluated as $x$ approaches 0.
3. The expression involves a square root, which suggests the possibility of rationalizing the denominator.

STEP 2

1. Identify the indeterminate form.
2. Rationalize the denominator.
3. Simplify the expression.
4. Evaluate the limit as $x$ approaches 0.

STEP 3

First, substitute $x = 0$ into the expression to check for indeterminate forms:
$\frac{0}{\sqrt{0+4} - 2} = \frac{0}{2 - 2} = \frac{0}{0}$
This is an indeterminate form.

STEP 4

To resolve the indeterminate form, rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator:
$\frac{x}{\sqrt{x+4} - 2} \times \frac{\sqrt{x+4} + 2}{\sqrt{x+4} + 2}$

STEP 5

Multiply out the denominator using the difference of squares formula:
$(\sqrt{x+4} - 2)(\sqrt{x+4} + 2) = (\sqrt{x+4})^2 - 2^2 = x + 4 - 4 = x$
The expression becomes:
$\frac{x(\sqrt{x+4} + 2)}{x}$

STEP 6

Simplify the expression by canceling $x$ in the numerator and the denominator:
$\sqrt{x+4} + 2$

STEP 7

Evaluate the limit as $x$ approaches 0:
$\lim_{x \rightarrow 0} (\sqrt{x+4} + 2) = \sqrt{0+4} + 2 = \sqrt{4} + 2 = 2 + 2 = 4$
The value of the limit is:
$\boxed{4}$

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Question4. Evaluale mathematically $\lim _{x \rightarrow 0} \frac{x}{\sqrt{x+4-2}}$

Studdy Solution

STEP 1

1. We are tasked with evaluating the limit $\lim_{x \rightarrow 0} \frac{x}{\sqrt{x+4} - 2}$ .
2. The limit is evaluated as $x$ approaches 0.
3. The expression involves a square root, which suggests the possibility of rationalizing the denominator.

STEP 2

1. Identify the indeterminate form.
2. Rationalize the denominator.
3. Simplify the expression.
4. Evaluate the limit as $x$ approaches 0.

STEP 3

First, substitute $x = 0$ into the expression to check for indeterminate forms:
$\frac{0}{\sqrt{0+4} - 2} = \frac{0}{2 - 2} = \frac{0}{0}$
This is an indeterminate form.

STEP 4

To resolve the indeterminate form, rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator:
$\frac{x}{\sqrt{x+4} - 2} \times \frac{\sqrt{x+4} + 2}{\sqrt{x+4} + 2}$

STEP 5

Multiply out the denominator using the difference of squares formula:
$(\sqrt{x+4} - 2)(\sqrt{x+4} + 2) = (\sqrt{x+4})^2 - 2^2 = x + 4 - 4 = x$
The expression becomes:
$\frac{x(\sqrt{x+4} + 2)}{x}$

STEP 6

Simplify the expression by canceling $x$ in the numerator and the denominator:
$\sqrt{x+4} + 2$

STEP 7

Evaluate the limit as $x$ approaches 0:
$\lim_{x \rightarrow 0} (\sqrt{x+4} + 2) = \sqrt{0+4} + 2 = \sqrt{4} + 2 = 2 + 2 = 4$
The value of the limit is:
$\boxed{4}$

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Question4. Evaluale mathematically lim⁡x→0xx+4−2\lim _{x \rightarrow 0} \frac{x}{\sqrt{x+4-2}}limx→0​x+4−2​x​

Studdy Solution

STEP 1

1. We are tasked with evaluating the limit lim⁡x→0xx+4−2\lim_{x \rightarrow 0} \frac{x}{\sqrt{x+4} - 2}limx→0​x+4​−2x​.2. The limit is evaluated as xxx approaches 0.3. The expression involves a square root, which suggests the possibility of rationalizing the denominator.

STEP 2

1. Identify the indeterminate form.2. Rationalize the denominator.3. Simplify the expression.4. Evaluate the limit as xxx approaches 0.

STEP 3

First, substitute x=0x = 0x=0 into the expression to check for indeterminate forms:00+4−2=02−2=00 \frac{0}{\sqrt{0+4} - 2} = \frac{0}{2 - 2} = \frac{0}{0} 0+4​−20​=2−20​=00​This is an indeterminate form.

STEP 4

To resolve the indeterminate form, rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator:xx+4−2×x+4+2x+4+2 \frac{x}{\sqrt{x+4} - 2} \times \frac{\sqrt{x+4} + 2}{\sqrt{x+4} + 2} x+4​−2x​×x+4​+2x+4​+2​

STEP 5

STEP 6

Simplify the expression by canceling xxx in the numerator and the denominator:x+4+2 \sqrt{x+4} + 2 x+4​+2

STEP 7

Evaluate the limit as xxx approaches 0:lim⁡x→0(x+4+2)=0+4+2=4+2=2+2=4 \lim_{x \rightarrow 0} (\sqrt{x+4} + 2) = \sqrt{0+4} + 2 = \sqrt{4} + 2 = 2 + 2 = 4 x→0lim​(x+4​+2)=0+4​+2=4​+2=2+2=4 The value of the limit is:4 \boxed{4} 4​

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Question4. Evaluale mathematically $\lim _{x \rightarrow 0} \frac{x}{\sqrt{x+4-2}}$

1. We are tasked with evaluating the limit $\lim_{x \rightarrow 0} \frac{x}{\sqrt{x+4} - 2}$ .
2. The limit is evaluated as $x$ approaches 0.
3. The expression involves a square root, which suggests the possibility of rationalizing the denominator.

1. Identify the indeterminate form.
2. Rationalize the denominator.
3. Simplify the expression.
4. Evaluate the limit as $x$ approaches 0.

First, substitute $x = 0$ into the expression to check for indeterminate forms:
$\frac{0}{\sqrt{0+4} - 2} = \frac{0}{2 - 2} = \frac{0}{0}$
This is an indeterminate form.

To resolve the indeterminate form, rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator:
$\frac{x}{\sqrt{x+4} - 2} \times \frac{\sqrt{x+4} + 2}{\sqrt{x+4} + 2}$

Simplify the expression by canceling $x$ in the numerator and the denominator:
$\sqrt{x+4} + 2$

Evaluate the limit as $x$ approaches 0:
$\lim_{x \rightarrow 0} (\sqrt{x+4} + 2) = \sqrt{0+4} + 2 = \sqrt{4} + 2 = 2 + 2 = 4$
The value of the limit is:
$\boxed{4}$