Calculus Examples

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

Step 1

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $7$ .

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

Step 3

Move the limit under the radical sign.

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

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

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

Step 5

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

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

Step 6

Evaluate the limits by plugging in $7$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

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

Step 6.2

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

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

Step 7

Simplify the answer.

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Step 7.1

Simplify the denominator.

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Step 7.1.1

Add $7$ and $2$ .

$3 \frac{7}{\sqrt{9}}$

Step 7.1.2

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

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

Step 7.1.3

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

$3 (\frac{7}{3})$

Step 7.2

Cancel the common factor of $3$ .

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Step 7.2.1

Cancel the common factor.

$3 (\frac{7}{3})$

Step 7.2.2

Rewrite the expression.

$7$