Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Move the exponent $3$ from ${(\frac{- 6 x + 1}{8})}^{3}$ outside the limit using the Limits Power Rule.

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

Step 1.2

Move the term $\frac{1}{8}$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 1.4

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.5

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

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

Step 2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

${(\frac{1}{8} (- 6 \cdot 1 + 1))}^{3}$

Step 3

Simplify the answer.

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Step 3.1

Multiply $- 6$ by $1$ .

${(\frac{1}{8} (- 6 + 1))}^{3}$

Step 3.2

Add $- 6$ and $1$ .

${(\frac{1}{8} \cdot - 5)}^{3}$

Step 3.3

Combine $\frac{1}{8}$ and $- 5$ .

${(\frac{- 5}{8})}^{3}$

Step 3.4

Move the negative in front of the fraction.

${(- \frac{5}{8})}^{3}$

Step 3.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.5.1

Apply the product rule to $- \frac{5}{8}$ .

${(- 1)}^{3} {(\frac{5}{8})}^{3}$

Step 3.5.2

Apply the product rule to $\frac{5}{8}$ .

${(- 1)}^{3} \frac{5^{3}}{8^{3}}$

Step 3.6

Raise $- 1$ to the power of $3$ .

$- \frac{5^{3}}{8^{3}}$

Step 3.7

Raise $5$ to the power of $3$ .

$- \frac{125}{8^{3}}$

Step 3.8

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

$- \frac{125}{512}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{125}{512}$

Decimal Form:

$- 0.24414062 \dots$