Calculus Examples

$\frac{lim_{x \to 8} 10 x^{3} - 9 x^{2} - 3 x}{8 - 4 x - 4 x^{3}}$

Step 1

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

$\frac{lim_{x \to 8} 10 x^{3} - lim_{x \to 8} 9 x^{2} - lim_{x \to 8} 3 x}{8 - 4 x - 4 x^{3}}$

Step 2

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\frac{10 lim_{x \to 8} x^{3} - lim_{x \to 8} 9 x^{2} - lim_{x \to 8} 3 x}{8 - 4 x - 4 x^{3}}$

Step 3

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

$\frac{10 {(lim_{x \to 8} x)}^{3} - lim_{x \to 8} 9 x^{2} - lim_{x \to 8} 3 x}{8 - 4 x - 4 x^{3}}$

Step 4

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{10 {(lim_{x \to 8} x)}^{3} - 9 lim_{x \to 8} x^{2} - lim_{x \to 8} 3 x}{8 - 4 x - 4 x^{3}}$

Step 5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{10 {(lim_{x \to 8} x)}^{3} - 9 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 3 x}{8 - 4 x - 4 x^{3}}$

Step 6

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

$\frac{10 {(lim_{x \to 8} x)}^{3} - 9 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x}{8 - 4 x - 4 x^{3}}$

Step 7

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

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Step 7.1

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

$\frac{10 \cdot 8^{3} - 9 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x}{8 - 4 x - 4 x^{3}}$

Step 7.2

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

$\frac{10 \cdot 8^{3} - 9 \cdot 8^{2} - 3 lim_{x \to 8} x}{8 - 4 x - 4 x^{3}}$

Step 7.3

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

$\frac{10 \cdot 8^{3} - 9 \cdot 8^{2} - 3 \cdot 8}{8 - 4 x - 4 x^{3}}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

$\frac{10 \cdot 512 - 9 \cdot 8^{2} - 3 \cdot 8}{8 - 4 x - 4 x^{3}}$

Step 8.1.2

Multiply $10$ by $512$ .

$\frac{5120 - 9 \cdot 8^{2} - 3 \cdot 8}{8 - 4 x - 4 x^{3}}$

Step 8.1.3

Raise $8$ to the power of $2$ .

$\frac{5120 - 9 \cdot 64 - 3 \cdot 8}{8 - 4 x - 4 x^{3}}$

Step 8.1.4

Multiply $- 9$ by $64$ .

$\frac{5120 - 576 - 3 \cdot 8}{8 - 4 x - 4 x^{3}}$

Step 8.1.5

Multiply $- 3$ by $8$ .

$\frac{5120 - 576 - 24}{8 - 4 x - 4 x^{3}}$

Step 8.1.6

Subtract $576$ from $5120$ .

$\frac{4544 - 24}{8 - 4 x - 4 x^{3}}$

Step 8.1.7

Subtract $24$ from $4544$ .

$\frac{4520}{8 - 4 x - 4 x^{3}}$

Step 8.2

Factor $4$ out of $8 - 4 x - 4 x^{3}$ .

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Step 8.2.1

Factor $4$ out of $8$ .

$\frac{4520}{4 (2) - 4 x - 4 x^{3}}$

Step 8.2.2

Factor $4$ out of $- 4 x$ .

$\frac{4520}{4 (2) + 4 (- x) - 4 x^{3}}$

Step 8.2.3

Factor $4$ out of $- 4 x^{3}$ .

$\frac{4520}{4 (2) + 4 (- x) + 4 (- x^{3})}$

Step 8.2.4

Factor $4$ out of $4 (2) + 4 (- x)$ .

$\frac{4520}{4 (2 - x) + 4 (- x^{3})}$

Step 8.2.5

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

$\frac{4520}{4 (2 - x - x^{3})}$

Step 8.3

Cancel the common factor of $4520$ and $4$ .

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Step 8.3.1

Factor $4$ out of $4520$ .

$\frac{4 \cdot 1130}{4 (2 - x - x^{3})}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Cancel the common factor.

$\frac{4 \cdot 1130}{4 (2 - x - x^{3})}$

Step 8.3.2.2

Rewrite the expression.

$\frac{1130}{2 - x - x^{3}}$