Calculus Examples

$lim_{y \to 2} {(\frac{4 y^{3} + 8 y}{y + 4})}^{\frac{1}{3}}$

Step 1

Move the exponent $\frac{1}{3}$ from ${(\frac{4 y^{3} + 8 y}{y + 4})}^{\frac{1}{3}}$ outside the limit using the Limits Power Rule.

${(lim_{y \to 2} \frac{4 y^{3} + 8 y}{y + 4})}^{\frac{1}{3}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $y$ approaches $2$ .

${(\frac{lim_{y \to 2} 4 y^{3} + 8 y}{lim_{y \to 2} y + 4})}^{\frac{1}{3}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $2$ .

${(\frac{lim_{y \to 2} 4 y^{3} + lim_{y \to 2} 8 y}{lim_{y \to 2} y + 4})}^{\frac{1}{3}}$

Step 4

Move the term $4$ outside of the limit because it is constant with respect to $y$ .

${(\frac{4 lim_{y \to 2} y^{3} + lim_{y \to 2} 8 y}{lim_{y \to 2} y + 4})}^{\frac{1}{3}}$

Step 5

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

${(\frac{4 {(lim_{y \to 2} y)}^{3} + lim_{y \to 2} 8 y}{lim_{y \to 2} y + 4})}^{\frac{1}{3}}$

Step 6

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

${(\frac{4 {(lim_{y \to 2} y)}^{3} + 8 lim_{y \to 2} y}{lim_{y \to 2} y + 4})}^{\frac{1}{3}}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $2$ .

${(\frac{4 {(lim_{y \to 2} y)}^{3} + 8 lim_{y \to 2} y}{lim_{y \to 2} y + lim_{y \to 2} 4})}^{\frac{1}{3}}$

Step 8

Evaluate the limit of $4$ which is constant as $y$ approaches $2$ .

${(\frac{4 {(lim_{y \to 2} y)}^{3} + 8 lim_{y \to 2} y}{lim_{y \to 2} y + 4})}^{\frac{1}{3}}$

Step 9

Evaluate the limits by plugging in $2$ for all occurrences of $y$ .

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Step 9.1

Evaluate the limit of $y$ by plugging in $2$ for $y$ .

${(\frac{4 \cdot 2^{3} + 8 lim_{y \to 2} y}{lim_{y \to 2} y + 4})}^{\frac{1}{3}}$

Step 9.2

Evaluate the limit of $y$ by plugging in $2$ for $y$ .

${(\frac{4 \cdot 2^{3} + 8 \cdot 2}{lim_{y \to 2} y + 4})}^{\frac{1}{3}}$

Step 9.3

Evaluate the limit of $y$ by plugging in $2$ for $y$ .

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

Step 10

Simplify the answer.

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Step 10.1

Cancel the common factor of $4 \cdot 2^{3} + 8 \cdot 2$ and $2 + 4$ .

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Step 10.1.1

Factor $2$ out of $4 \cdot 2^{3}$ .

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

Step 10.1.2

Factor $2$ out of $8 \cdot 2$ .

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

Step 10.1.3

Factor $2$ out of $2 (2 \cdot 2^{3}) + 2 (4 \cdot 2)$ .

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

Step 10.1.4

Cancel the common factors.

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Step 10.1.4.1

Factor $2$ out of $2$ .

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

Step 10.1.4.2

Factor $2$ out of $4$ .

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

Step 10.1.4.3

Factor $2$ out of $2 \cdot 1 + 2 \cdot 2$ .

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

Step 10.1.4.4

Cancel the common factor.

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

Step 10.1.4.5

Rewrite the expression.

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

Step 10.2

Simplify the numerator.

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Step 10.2.1

Multiply $2$ by $2^{3}$ by adding the exponents.

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Step 10.2.1.1

Multiply $2$ by $2^{3}$ .

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Step 10.2.1.1.1

Raise $2$ to the power of $1$ .

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

Step 10.2.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 10.2.1.2

Add $1$ and $3$ .

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

Step 10.2.2

Raise $2$ to the power of $4$ .

${(\frac{16 + 4 \cdot 2}{1 + 2})}^{\frac{1}{3}}$

Step 10.2.3

Multiply $4$ by $2$ .

${(\frac{16 + 8}{1 + 2})}^{\frac{1}{3}}$

Step 10.2.4

Add $16$ and $8$ .

${(\frac{24}{1 + 2})}^{\frac{1}{3}}$

Step 10.3

Add $1$ and $2$ .

${(\frac{24}{3})}^{\frac{1}{3}}$

Step 10.4

Divide $24$ by $3$ .

$8^{\frac{1}{3}}$

Step 10.5

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

${(2^{3})}^{\frac{1}{3}}$

Step 10.6

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

$2^{3 (\frac{1}{3})}$

Step 10.7

Cancel the common factor of $3$ .

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Step 10.7.1

Cancel the common factor.

$2^{3 (\frac{1}{3})}$

Step 10.7.2

Rewrite the expression.

$2^{1}$

Step 10.8

Evaluate the exponent.

$2$