Calculus Examples

$lim_{x \to 4} \frac{12 + \sqrt{x}}{\sqrt{12 + x}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} 12 + \sqrt{x}}{lim_{x \to 4} \sqrt{12 + x}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} 12 + lim_{x \to 4} \sqrt{x}}{lim_{x \to 4} \sqrt{12 + x}}$

Step 3

Evaluate the limit of $12$ which is constant as $x$ approaches $4$ .

$\frac{12 + lim_{x \to 4} \sqrt{x}}{lim_{x \to 4} \sqrt{12 + x}}$

Step 4

Move the limit under the radical sign.

$\frac{12 + \sqrt{lim_{x \to 4} x}}{lim_{x \to 4} \sqrt{12 + x}}$

Step 5

Move the limit under the radical sign.

$\frac{12 + \sqrt{lim_{x \to 4} x}}{\sqrt{lim_{x \to 4} 12 + x}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{12 + \sqrt{lim_{x \to 4} x}}{\sqrt{lim_{x \to 4} 12 + lim_{x \to 4} x}}$

Step 7

Evaluate the limit of $12$ which is constant as $x$ approaches $4$ .

$\frac{12 + \sqrt{lim_{x \to 4} x}}{\sqrt{12 + lim_{x \to 4} x}}$

Step 8

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{12 + \sqrt{4}}{\sqrt{12 + lim_{x \to 4} x}}$

Step 8.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{12 + \sqrt{4}}{\sqrt{12 + 4}}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Rewrite $4$ as $2^{2}$ .

$\frac{12 + \sqrt{2^{2}}}{\sqrt{12 + 4}}$

Step 9.1.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{12 + 2}{\sqrt{12 + 4}}$

Step 9.1.3

Add $12$ and $2$ .

$\frac{14}{\sqrt{12 + 4}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Add $12$ and $4$ .

$\frac{14}{\sqrt{16}}$

Step 9.2.2

Rewrite $16$ as $4^{2}$ .

$\frac{14}{\sqrt{4^{2}}}$

Step 9.2.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{14}{4}$

Step 9.3

Cancel the common factor of $14$ and $4$ .

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Step 9.3.1

Factor $2$ out of $14$ .

$\frac{2 (7)}{4}$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 7}{2 \cdot 2}$

Step 9.3.2.2

Cancel the common factor.

$\frac{2 \cdot 7}{2 \cdot 2}$

Step 9.3.2.3

Rewrite the expression.

$\frac{7}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{2}$

Decimal Form:

$3.5$