Calculus Examples

$lim_{x \to 4} 2^{x} + \sqrt{x + 5}$

Step 1

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

$lim_{x \to 4} 2^{x} + lim_{x \to 4} \sqrt{x + 5}$

Step 2

Move the limit into the exponent.

$2^{lim_{x \to 4} x} + lim_{x \to 4} \sqrt{x + 5}$

Step 3

Move the limit under the radical sign.

$2^{lim_{x \to 4} x} + \sqrt{lim_{x \to 4} x + 5}$

Step 4

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

$2^{lim_{x \to 4} x} + \sqrt{lim_{x \to 4} x + lim_{x \to 4} 5}$

Step 5

Evaluate the limit of $5$ which is constant as $x$ approaches $4$ .

$2^{lim_{x \to 4} x} + \sqrt{lim_{x \to 4} x + 5}$

Step 6

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

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Step 6.1

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

$2^{4} + \sqrt{lim_{x \to 4} x + 5}$

Step 6.2

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

$2^{4} + \sqrt{4 + 5}$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Raise $2$ to the power of $4$ .

$16 + \sqrt{4 + 5}$

Step 7.1.2

Add $4$ and $5$ .

$16 + \sqrt{9}$

Step 7.1.3

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

$16 + \sqrt{3^{2}}$

Step 7.1.4

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

$16 + 3$

Step 7.2

Add $16$ and $3$ .

$19$