Calculus Examples

$lim_{x \to 2} \frac{\sqrt{x + 7} + 3}{x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} \sqrt{x + 7} + 3}{lim_{x \to 2} x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} \sqrt{x + 7} + lim_{x \to 2} 3}{lim_{x \to 2} x}$

Step 3

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 2} x + 7} + lim_{x \to 2} 3}{lim_{x \to 2} x}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{\sqrt{lim_{x \to 2} x + lim_{x \to 2} 7} + lim_{x \to 2} 3}{lim_{x \to 2} x}$

Step 5

Evaluate the limit of $7$ which is constant as $x$ approaches $2$ .

$\frac{\sqrt{lim_{x \to 2} x + 7} + lim_{x \to 2} 3}{lim_{x \to 2} x}$

Step 6

Evaluate the limit of $3$ which is constant as $x$ approaches $2$ .

$\frac{\sqrt{lim_{x \to 2} x + 7} + 3}{lim_{x \to 2} x}$

Step 7

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{\sqrt{2 + 7} + 3}{lim_{x \to 2} x}$

Step 7.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{\sqrt{2 + 7} + 3}{2}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Add $2$ and $7$ .

$\frac{\sqrt{9} + 3}{2}$

Step 8.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2}} + 3}{2}$

Step 8.1.3

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

$\frac{3 + 3}{2}$

Step 8.1.4

Add $3$ and $3$ .

$\frac{6}{2}$

Step 8.2

Divide $6$ by $2$ .

$3$