Calculus Examples

$\frac{lim_{x \to 8} \sqrt{3 x^{12} - 16}}{x^{6} - 3}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} 3 x^{12} - 16}}{x^{6} - 3}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{\sqrt{lim_{x \to 8} 3 x^{12} - lim_{x \to 8} 16}}{x^{6} - 3}$

Step 1.3

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

$\frac{\sqrt{3 lim_{x \to 8} x^{12} - lim_{x \to 8} 16}}{x^{6} - 3}$

Step 1.4

Move the exponent $12$ from $x^{12}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{3 {(lim_{x \to 8} x)}^{12} - lim_{x \to 8} 16}}{x^{6} - 3}$

Step 1.5

Evaluate the limit of $16$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{3 {(lim_{x \to 8} x)}^{12} - 1 \cdot 16}}{x^{6} - 3}$

Step 2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{\sqrt{3 \cdot 8^{12} - 1 \cdot 16}}{x^{6} - 3}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $8$ to the power of $12$ .

$\frac{\sqrt{3 \cdot 68719476736 - 1 \cdot 16}}{x^{6} - 3}$

Step 3.2

Multiply $3$ by $68719476736$ .

$\frac{\sqrt{206158430208 - 1 \cdot 16}}{x^{6} - 3}$

Step 3.3

Multiply $- 1$ by $16$ .

$\frac{\sqrt{206158430208 - 16}}{x^{6} - 3}$

Step 3.4

Subtract $16$ from $206158430208$ .

$\frac{\sqrt{206158430192}}{x^{6} - 3}$

Step 3.5

Rewrite $206158430192$ as $52^{2} \cdot 76242023$ .

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Step 3.5.1

Factor $2704$ out of $206158430192$ .

$\frac{\sqrt{2704 (76242023)}}{x^{6} - 3}$

Step 3.5.2

Rewrite $2704$ as $52^{2}$ .

$\frac{\sqrt{52^{2} \cdot 76242023}}{x^{6} - 3}$

Step 3.6

Pull terms out from under the radical.

$\frac{52 \sqrt{76242023}}{x^{6} - 3}$