Calculus Examples

$lim_{x \to 8} x^{\frac{1}{3}} (x - 4)$

Step 1

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

$lim_{x \to 8} x^{\frac{1}{3}} \cdot lim_{x \to 8} x - 4$

Step 2

Move the exponent $\frac{1}{3}$ from $x^{\frac{1}{3}}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 8} x)}^{\frac{1}{3}} \cdot lim_{x \to 8} x - 4$

Step 3

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

${(lim_{x \to 8} x)}^{\frac{1}{3}} \cdot (lim_{x \to 8} x - lim_{x \to 8} 4)$

Step 4

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

${(lim_{x \to 8} x)}^{\frac{1}{3}} \cdot (lim_{x \to 8} x - 1 \cdot 4)$

Step 5

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

Tap for more steps...

Step 5.1

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

$8^{\frac{1}{3}} \cdot (lim_{x \to 8} x - 1 \cdot 4)$

Step 5.2

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

$8^{\frac{1}{3}} \cdot (8 - 1 \cdot 4)$

Step 6

Simplify the answer.

Tap for more steps...

Step 6.1

Rewrite $8$ as $2^{3}$ .

${(2^{3})}^{\frac{1}{3}} \cdot (8 - 1 \cdot 4)$

Step 6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2^{3 (\frac{1}{3})} \cdot (8 - 1 \cdot 4)$

Step 6.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.3.1

Cancel the common factor.

$2^{3 (\frac{1}{3})} \cdot (8 - 1 \cdot 4)$

Step 6.3.2

Rewrite the expression.

$2^{1} \cdot (8 - 1 \cdot 4)$

Step 6.4

Evaluate the exponent.

$2 \cdot (8 - 1 \cdot 4)$

Step 6.5

Multiply $- 1$ by $4$ .

$2 \cdot (8 - 4)$

Step 6.6

Subtract $4$ from $8$ .

$2 \cdot 4$

Step 6.7

Multiply $2$ by $4$ .

$8$