Calculus Examples

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

Step 1

Move the exponent $\frac{1}{3}$ from ${(\frac{x^{2} + x - 1}{8 x^{2} - 3})}^{\frac{1}{3}}$ outside the limit using the Limits Power Rule.

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

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 8$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $- 8$ .

${(\frac{{(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} x - 1 \cdot 1}{lim_{x \to - 8} 8 x^{2} - 3})}^{\frac{1}{3}}$

Step 6

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

${(\frac{{(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} x - 1 \cdot 1}{lim_{x \to - 8} 8 x^{2} - lim_{x \to - 8} 3})}^{\frac{1}{3}}$

Step 7

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

${(\frac{{(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} x - 1 \cdot 1}{8 lim_{x \to - 8} x^{2} - lim_{x \to - 8} 3})}^{\frac{1}{3}}$

Step 8

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

${(\frac{{(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} x - 1 \cdot 1}{8 {(lim_{x \to - 8} x)}^{2} - lim_{x \to - 8} 3})}^{\frac{1}{3}}$

Step 9

Evaluate the limit of $3$ which is constant as $x$ approaches $- 8$ .

${(\frac{{(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} x - 1 \cdot 1}{8 {(lim_{x \to - 8} x)}^{2} - 1 \cdot 3})}^{\frac{1}{3}}$

Step 10

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

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Step 10.1

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

${(\frac{{(- 8)}^{2} + lim_{x \to - 8} x - 1 \cdot 1}{8 {(lim_{x \to - 8} x)}^{2} - 1 \cdot 3})}^{\frac{1}{3}}$

Step 10.2

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

${(\frac{{(- 8)}^{2} - 8 - 1 \cdot 1}{8 {(lim_{x \to - 8} x)}^{2} - 1 \cdot 3})}^{\frac{1}{3}}$

Step 10.3

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

${(\frac{{(- 8)}^{2} - 8 - 1 \cdot 1}{8 {(- 8)}^{2} - 1 \cdot 3})}^{\frac{1}{3}}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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

${(\frac{64 - 8 - 1 \cdot 1}{8 {(- 8)}^{2} - 1 \cdot 3})}^{\frac{1}{3}}$

Step 11.1.2

Multiply $- 1$ by $1$ .

${(\frac{64 - 8 - 1}{8 {(- 8)}^{2} - 1 \cdot 3})}^{\frac{1}{3}}$

Step 11.1.3

Subtract $8$ from $64$ .

${(\frac{56 - 1}{8 {(- 8)}^{2} - 1 \cdot 3})}^{\frac{1}{3}}$

Step 11.1.4

Subtract $1$ from $56$ .

${(\frac{55}{8 {(- 8)}^{2} - 1 \cdot 3})}^{\frac{1}{3}}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

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

${(\frac{55}{8 \cdot 64 - 1 \cdot 3})}^{\frac{1}{3}}$

Step 11.2.2

Multiply $8$ by $64$ .

${(\frac{55}{512 - 1 \cdot 3})}^{\frac{1}{3}}$

Step 11.2.3

Multiply $- 1$ by $3$ .

${(\frac{55}{512 - 3})}^{\frac{1}{3}}$

Step 11.2.4

Subtract $3$ from $512$ .

${(\frac{55}{509})}^{\frac{1}{3}}$

Step 11.3

Apply the product rule to $\frac{55}{509}$ .

$\frac{55^{\frac{1}{3}}}{509^{\frac{1}{3}}}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{55^{\frac{1}{3}}}{509^{\frac{1}{3}}}$

Decimal Form:

$0.47630115 \dots$