Calculus Examples

$\frac{lim_{x \to 8 x} 5 x^{5} + 2 x^{3} + 6}{- 6 x^{6} + 2 x^{4}}$

Step 1

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

$\frac{lim_{x \to 8 x} 5 x^{5} + lim_{x \to 8 x} 2 x^{3} + lim_{x \to 8 x} 6}{- 6 x^{6} + 2 x^{4}}$

Step 2

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 lim_{x \to 8 x} x^{5} + lim_{x \to 8 x} 2 x^{3} + lim_{x \to 8 x} 6}{- 6 x^{6} + 2 x^{4}}$

Step 3

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$\frac{5 {(lim_{x \to 8 x} x)}^{5} + lim_{x \to 8 x} 2 x^{3} + lim_{x \to 8 x} 6}{- 6 x^{6} + 2 x^{4}}$

Step 4

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

$\frac{5 {(lim_{x \to 8 x} x)}^{5} + 2 lim_{x \to 8 x} x^{3} + lim_{x \to 8 x} 6}{- 6 x^{6} + 2 x^{4}}$

Step 5

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

$\frac{5 {(lim_{x \to 8 x} x)}^{5} + 2 {(lim_{x \to 8 x} x)}^{3} + lim_{x \to 8 x} 6}{- 6 x^{6} + 2 x^{4}}$

Step 6

Evaluate the limit of $6$ which is constant as $x$ approaches $8 x$ .

$\frac{5 {(lim_{x \to 8 x} x)}^{5} + 2 {(lim_{x \to 8 x} x)}^{3} + 6}{- 6 x^{6} + 2 x^{4}}$

Step 7

Evaluate the limits by plugging in $8 x$ for all occurrences of $x$ .

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Step 7.1

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

$\frac{5 {(8 x)}^{5} + 2 {(lim_{x \to 8 x} x)}^{3} + 6}{- 6 x^{6} + 2 x^{4}}$

Step 7.2

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

$\frac{5 {(8 x)}^{5} + 2 {(8 x)}^{3} + 6}{- 6 x^{6} + 2 x^{4}}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Apply the product rule to $8 x$ .

$\frac{5 (8^{5} x^{5}) + 2 {(8 x)}^{3} + 6}{- 6 x^{6} + 2 x^{4}}$

Step 8.1.2

Raise $8$ to the power of $5$ .

$\frac{5 (32768 x^{5}) + 2 {(8 x)}^{3} + 6}{- 6 x^{6} + 2 x^{4}}$

Step 8.1.3

Multiply $32768$ by $5$ .

$\frac{163840 x^{5} + 2 {(8 x)}^{3} + 6}{- 6 x^{6} + 2 x^{4}}$

Step 8.1.4

Apply the product rule to $8 x$ .

$\frac{163840 x^{5} + 2 (8^{3} x^{3}) + 6}{- 6 x^{6} + 2 x^{4}}$

Step 8.1.5

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

$\frac{163840 x^{5} + 2 (512 x^{3}) + 6}{- 6 x^{6} + 2 x^{4}}$

Step 8.1.6

Multiply $512$ by $2$ .

$\frac{163840 x^{5} + 1024 x^{3} + 6}{- 6 x^{6} + 2 x^{4}}$

Step 8.1.7

Factor $2$ out of $163840 x^{5} + 1024 x^{3} + 6$ .

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Step 8.1.7.1

Factor $2$ out of $163840 x^{5}$ .

$\frac{2 (81920 x^{5}) + 1024 x^{3} + 6}{- 6 x^{6} + 2 x^{4}}$

Step 8.1.7.2

Factor $2$ out of $1024 x^{3}$ .

$\frac{2 (81920 x^{5}) + 2 (512 x^{3}) + 6}{- 6 x^{6} + 2 x^{4}}$

Step 8.1.7.3

Factor $2$ out of $6$ .

$\frac{2 (81920 x^{5}) + 2 (512 x^{3}) + 2 (3)}{- 6 x^{6} + 2 x^{4}}$

Step 8.1.7.4

Factor $2$ out of $2 (81920 x^{5}) + 2 (512 x^{3})$ .

$\frac{2 (81920 x^{5} + 512 x^{3}) + 2 (3)}{- 6 x^{6} + 2 x^{4}}$

Step 8.1.7.5

Factor $2$ out of $2 (81920 x^{5} + 512 x^{3}) + 2 (3)$ .

$\frac{2 (81920 x^{5} + 512 x^{3} + 3)}{- 6 x^{6} + 2 x^{4}}$

Step 8.2

Factor $2 x^{4}$ out of $- 6 x^{6} + 2 x^{4}$ .

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Step 8.2.1

Factor $2 x^{4}$ out of $- 6 x^{6}$ .

$\frac{2 (81920 x^{5} + 512 x^{3} + 3)}{2 x^{4} (- 3 x^{2}) + 2 x^{4}}$

Step 8.2.2

Factor $2 x^{4}$ out of $2 x^{4}$ .

$\frac{2 (81920 x^{5} + 512 x^{3} + 3)}{2 x^{4} (- 3 x^{2}) + 2 x^{4} (1)}$

Step 8.2.3

Factor $2 x^{4}$ out of $2 x^{4} (- 3 x^{2}) + 2 x^{4} (1)$ .

$\frac{2 (81920 x^{5} + 512 x^{3} + 3)}{2 x^{4} (- 3 x^{2} + 1)}$

Step 8.3

Cancel the common factor of $2$ .

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Step 8.3.1

Cancel the common factor.

$\frac{2 (81920 x^{5} + 512 x^{3} + 3)}{2 x^{4} (- 3 x^{2} + 1)}$

Step 8.3.2

Rewrite the expression.

$\frac{81920 x^{5} + 512 x^{3} + 3}{x^{4} (- 3 x^{2} + 1)}$

Step 8.4

Factor $- 1$ out of $- 3 x^{2}$ .

$\frac{81920 x^{5} + 512 x^{3} + 3}{x^{4} (- (3 x^{2}) + 1)}$

Step 8.5

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

$\frac{81920 x^{5} + 512 x^{3} + 3}{x^{4} (- (3 x^{2}) - 1 (- 1))}$

Step 8.6

Factor $- 1$ out of $- (3 x^{2}) - 1 (- 1)$ .

$\frac{81920 x^{5} + 512 x^{3} + 3}{x^{4} (- (3 x^{2} - 1))}$

Step 8.7

Rewrite $- (3 x^{2} - 1)$ as $- 1 (3 x^{2} - 1)$ .

$\frac{81920 x^{5} + 512 x^{3} + 3}{x^{4} (- 1 (3 x^{2} - 1))}$

Step 8.8

Move the negative in front of the fraction.

$- \frac{81920 x^{5} + 512 x^{3} + 3}{x^{4} (3 x^{2} - 1)}$