Calculus Examples

$lim_{x \to 0} \sqrt{7 + 2 e^{2 x}}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 0} 7 + 2 e^{2 x}}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\sqrt{lim_{x \to 0} 7 + lim_{x \to 0} 2 e^{2 x}}$

Step 1.3

Evaluate the limit of $7$ which is constant as $x$ approaches $0$ .

$\sqrt{7 + lim_{x \to 0} 2 e^{2 x}}$

Step 1.4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{7 + 2 lim_{x \to 0} e^{2 x}}$

Step 1.5

Move the limit into the exponent.

$\sqrt{7 + 2 e^{lim_{x \to 0} 2 x}}$

Step 1.6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{7 + 2 e^{2 lim_{x \to 0} x}}$

Step 2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\sqrt{7 + 2 e^{2 \cdot 0}}$

Step 3

Simplify the answer.

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Step 3.1

Multiply $2$ by $0$ .

$\sqrt{7 + 2 e^{0}}$

Step 3.2

Anything raised to $0$ is $1$ .

$\sqrt{7 + 2 \cdot 1}$

Step 3.3

Multiply $2$ by $1$ .

$\sqrt{7 + 2}$

Step 3.4

Add $7$ and $2$ .

$\sqrt{9}$

Step 3.5

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

$\sqrt{3^{2}}$

Step 3.6

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

$3$