Calculus Examples

$\frac{lim_{x \to 64} \sqrt{x} - 8}{x - 64}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $64$ .

$\frac{lim_{x \to 64} \sqrt{x} - lim_{x \to 64} 8}{x - 64}$

Step 1.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 64} x} - lim_{x \to 64} 8}{x - 64}$

Step 1.3

Evaluate the limit of $8$ which is constant as $x$ approaches $64$ .

$\frac{\sqrt{lim_{x \to 64} x} - 1 \cdot 8}{x - 64}$

Step 2

Evaluate the limit of $x$ by plugging in $64$ for $x$ .

$\frac{\sqrt{64} - 1 \cdot 8}{x - 64}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Rewrite $64$ as $8^{2}$ .

$\frac{\sqrt{8^{2}} - 1 \cdot 8}{x - 64}$

Step 3.1.2

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

$\frac{8 - 1 \cdot 8}{x - 64}$

Step 3.1.3

Multiply $- 1$ by $8$ .

$\frac{8 - 8}{x - 64}$

Step 3.1.4

Subtract $8$ from $8$ .

$\frac{0}{x - 64}$

Step 3.2

Divide $0$ by $x - 64$ .

$0$