Calculus Examples

$\frac{lim_{x \to 8} 4 x^{3} + 2}{x^{3} + \sqrt{4 x^{6} + 1}}$

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 $8$ .

$\frac{lim_{x \to 8} 4 x^{3} + lim_{x \to 8} 2}{x^{3} + \sqrt{4 x^{6} + 1}}$

Step 1.2

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to 8} x^{3} + lim_{x \to 8} 2}{x^{3} + \sqrt{4 x^{6} + 1}}$

Step 1.3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{4 {(lim_{x \to 8} x)}^{3} + lim_{x \to 8} 2}{x^{3} + \sqrt{4 x^{6} + 1}}$

Step 1.4

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{4 {(lim_{x \to 8} x)}^{3} + 2}{x^{3} + \sqrt{4 x^{6} + 1}}$

Step 2

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

$\frac{4 \cdot 8^{3} + 2}{x^{3} + \sqrt{4 x^{6} + 1}}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $8$ to the power of $3$ .

$\frac{4 \cdot 512 + 2}{x^{3} + \sqrt{4 x^{6} + 1}}$

Step 3.1.2

Multiply $4$ by $512$ .

$\frac{2048 + 2}{x^{3} + \sqrt{4 x^{6} + 1}}$

Step 3.1.3

Add $2048$ and $2$ .

$\frac{2050}{x^{3} + \sqrt{4 x^{6} + 1}}$

Step 3.2

Multiply $\frac{2050}{x^{3} + \sqrt{4 x^{6} + 1}}$ by $\frac{x^{3} - \sqrt{4 x^{6} + 1}}{x^{3} - \sqrt{4 x^{6} + 1}}$ .

$\frac{2050}{x^{3} + \sqrt{4 x^{6} + 1}} \cdot \frac{x^{3} - \sqrt{4 x^{6} + 1}}{x^{3} - \sqrt{4 x^{6} + 1}}$

Step 3.3

Multiply $\frac{2050}{x^{3} + \sqrt{4 x^{6} + 1}}$ by $\frac{x^{3} - \sqrt{4 x^{6} + 1}}{x^{3} - \sqrt{4 x^{6} + 1}}$ .

$\frac{2050 (x^{3} - \sqrt{4 x^{6} + 1})}{(x^{3} + \sqrt{4 x^{6} + 1}) (x^{3} - \sqrt{4 x^{6} + 1})}$

Step 3.4

Expand the denominator using the FOIL method.

$\frac{2050 (x^{3} - \sqrt{4 x^{6} + 1})}{x^{6} + x^{3} (- \sqrt{4 x^{6} + 1}) + \sqrt{4 x^{6} + 1} x^{3} - {\sqrt{4 x^{6} + 1}}^{2}}$

Step 3.5

Simplify.

$\frac{2050 (x^{3} - \sqrt{4 x^{6} + 1})}{- 3 x^{6} - 1}$

Step 3.6

Factor $- 1$ out of $- 3 x^{6}$ .

$\frac{2050 (x^{3} - \sqrt{4 x^{6} + 1})}{- (3 x^{6}) - 1}$

Step 3.7

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{2050 (x^{3} - \sqrt{4 x^{6} + 1})}{- (3 x^{6}) - 1 (1)}$

Step 3.8

Factor $- 1$ out of $- (3 x^{6}) - 1 (1)$ .

$\frac{2050 (x^{3} - \sqrt{4 x^{6} + 1})}{- (3 x^{6} + 1)}$

Step 3.9

Rewrite $- (3 x^{6} + 1)$ as $- 1 (3 x^{6} + 1)$ .

$\frac{2050 (x^{3} - \sqrt{4 x^{6} + 1})}{- 1 (3 x^{6} + 1)}$

Step 3.10

Move the negative in front of the fraction.

$- \frac{2050 (x^{3} - \sqrt{4 x^{6} + 1})}{3 x^{6} + 1}$