Calculus Examples

$lim_{x \to 3} \sqrt[3]{\frac{2 + 5 x - 3 x^{4}}{x^{2} - 1}}$

Step 1

Move the limit under the radical sign.

$\sqrt[3]{lim_{x \to 3} \frac{2 + 5 x - 3 x^{4}}{x^{2} - 1}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\sqrt[3]{\frac{lim_{x \to 3} 2 + 5 x - 3 x^{4}}{lim_{x \to 3} x^{2} - 1}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\sqrt[3]{\frac{lim_{x \to 3} 2 + lim_{x \to 3} 5 x - lim_{x \to 3} 3 x^{4}}{lim_{x \to 3} x^{2} - 1}}$

Step 4

Evaluate the limit of $2$ which is constant as $x$ approaches $3$ .

$\sqrt[3]{\frac{2 + lim_{x \to 3} 5 x - lim_{x \to 3} 3 x^{4}}{lim_{x \to 3} x^{2} - 1}}$

Step 5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\sqrt[3]{\frac{2 + 5 lim_{x \to 3} x - lim_{x \to 3} 3 x^{4}}{lim_{x \to 3} x^{2} - 1}}$

Step 6

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

$\sqrt[3]{\frac{2 + 5 lim_{x \to 3} x - 3 lim_{x \to 3} x^{4}}{lim_{x \to 3} x^{2} - 1}}$

Step 7

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\sqrt[3]{\frac{2 + 5 lim_{x \to 3} x - 3 {(lim_{x \to 3} x)}^{4}}{lim_{x \to 3} x^{2} - 1}}$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\sqrt[3]{\frac{2 + 5 lim_{x \to 3} x - 3 {(lim_{x \to 3} x)}^{4}}{lim_{x \to 3} x^{2} - lim_{x \to 3} 1}}$

Step 9

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

$\sqrt[3]{\frac{2 + 5 lim_{x \to 3} x - 3 {(lim_{x \to 3} x)}^{4}}{{(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 1}}$

Step 10

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

$\sqrt[3]{\frac{2 + 5 lim_{x \to 3} x - 3 {(lim_{x \to 3} x)}^{4}}{{(lim_{x \to 3} x)}^{2} - 1 \cdot 1}}$

Step 11

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

Tap for more steps...

Step 11.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\sqrt[3]{\frac{2 + 5 \cdot 3 - 3 {(lim_{x \to 3} x)}^{4}}{{(lim_{x \to 3} x)}^{2} - 1 \cdot 1}}$

Step 11.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\sqrt[3]{\frac{2 + 5 \cdot 3 - 3 \cdot 3^{4}}{{(lim_{x \to 3} x)}^{2} - 1 \cdot 1}}$

Step 11.3

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\sqrt[3]{\frac{2 + 5 \cdot 3 - 3 \cdot 3^{4}}{3^{2} - 1 \cdot 1}}$

Step 12

Simplify the answer.

Tap for more steps...

Step 12.1

Multiply $5$ by $3$ .

$\sqrt[3]{\frac{2 + 15 - 3 \cdot 3^{4}}{3^{2} - 1 \cdot 1}}$

Step 12.2

Raise $3$ to the power of $4$ .

$\sqrt[3]{\frac{2 + 15 - 3 \cdot 81}{3^{2} - 1 \cdot 1}}$

Step 12.3

Multiply $- 3$ by $81$ .

$\sqrt[3]{\frac{2 + 15 - 243}{3^{2} - 1 \cdot 1}}$

Step 12.4

Add $2$ and $15$ .

$\sqrt[3]{\frac{17 - 243}{3^{2} - 1 \cdot 1}}$

Step 12.5

Subtract $243$ from $17$ .

$\sqrt[3]{\frac{- 226}{3^{2} - 1 \cdot 1}}$

Step 12.6

Raise $3$ to the power of $2$ .

$\sqrt[3]{\frac{- 226}{9 - 1 \cdot 1}}$

Step 12.7

Multiply $- 1$ by $1$ .

$\sqrt[3]{\frac{- 226}{9 - 1}}$

Step 12.8

Subtract $1$ from $9$ .

$\sqrt[3]{\frac{- 226}{8}}$

Step 12.9

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 12.9.1

Cancel the common factor of $- 226$ and $8$ .

Tap for more steps...

Step 12.9.1.1

Factor $2$ out of $- 226$ .

$\sqrt[3]{\frac{2 (- 113)}{8}}$

Step 12.9.1.2

Cancel the common factors.

Tap for more steps...

Step 12.9.1.2.1

Factor $2$ out of $8$ .

$\sqrt[3]{\frac{2 \cdot - 113}{2 \cdot 4}}$

Step 12.9.1.2.2

Cancel the common factor.

$\sqrt[3]{\frac{2 \cdot - 113}{2 \cdot 4}}$

Step 12.9.1.2.3

Rewrite the expression.

$\sqrt[3]{\frac{- 113}{4}}$

Step 12.9.2

Move the negative in front of the fraction.

$\sqrt[3]{- \frac{113}{4}}$

Step 12.10

Rewrite $- \frac{113}{4}$ as ${({(- 1)}^{3})}^{3} \frac{113}{4}$ .

Tap for more steps...

Step 12.10.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \frac{113}{4}}$

Step 12.10.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\sqrt[3]{{({(- 1)}^{3})}^{3} \frac{113}{4}}$

Step 12.11

Pull terms out from under the radical.

${(- 1)}^{3} \sqrt[3]{\frac{113}{4}}$

Step 12.12

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

$- \sqrt[3]{\frac{113}{4}}$

Step 12.13

Rewrite $\sqrt[3]{\frac{113}{4}}$ as $\frac{\sqrt[3]{113}}{\sqrt[3]{4}}$ .

$- \frac{\sqrt[3]{113}}{\sqrt[3]{4}}$

Step 12.14

Multiply $\frac{\sqrt[3]{113}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$- (\frac{\sqrt[3]{113}}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}})$

Step 12.15

Combine and simplify the denominator.

Tap for more steps...

Step 12.15.1

Multiply $\frac{\sqrt[3]{113}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$- \frac{\sqrt[3]{113} {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 12.15.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$- \frac{\sqrt[3]{113} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

Step 12.15.3

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

$- \frac{\sqrt[3]{113} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 12.15.4

Add $1$ and $2$ .

$- \frac{\sqrt[3]{113} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Step 12.15.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

Tap for more steps...

Step 12.15.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$- \frac{\sqrt[3]{113} {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

Step 12.15.5.2

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

$- \frac{\sqrt[3]{113} {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

Step 12.15.5.3

Combine $\frac{1}{3}$ and $3$ .

$- \frac{\sqrt[3]{113} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 12.15.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 12.15.5.4.1

Cancel the common factor.

$- \frac{\sqrt[3]{113} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 12.15.5.4.2

Rewrite the expression.

$- \frac{\sqrt[3]{113} {\sqrt[3]{4}}^{2}}{4^{1}}$

Step 12.15.5.5

Evaluate the exponent.

$- \frac{\sqrt[3]{113} {\sqrt[3]{4}}^{2}}{4}$

Step 12.16

Simplify the numerator.

Tap for more steps...

Step 12.16.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$- \frac{\sqrt[3]{113} \sqrt[3]{4^{2}}}{4}$

Step 12.16.2

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

$- \frac{\sqrt[3]{113} \sqrt[3]{16}}{4}$

Step 12.16.3

Rewrite $16$ as $2^{3} \cdot 2$ .

Tap for more steps...

Step 12.16.3.1

Factor $8$ out of $16$ .

$- \frac{\sqrt[3]{113} \sqrt[3]{8 (2)}}{4}$

Step 12.16.3.2

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

$- \frac{\sqrt[3]{113} \sqrt[3]{2^{3} \cdot 2}}{4}$

Step 12.16.4

Pull terms out from under the radical.

$- \frac{\sqrt[3]{113} \cdot 2 \sqrt[3]{2}}{4}$

Step 12.16.5

Combine exponents.

Tap for more steps...

Step 12.16.5.1

Combine using the product rule for radicals.

$- \frac{2 \sqrt[3]{113 \cdot 2}}{4}$

Step 12.16.5.2

Multiply $113$ by $2$ .

$- \frac{2 \sqrt[3]{226}}{4}$

Step 12.17

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 12.17.1

Factor $2$ out of $2 \sqrt[3]{226}$ .

$- \frac{2 (\sqrt[3]{226})}{4}$

Step 12.17.2

Cancel the common factors.

Tap for more steps...

Step 12.17.2.1

Factor $2$ out of $4$ .

$- \frac{2 \sqrt[3]{226}}{2 \cdot 2}$

Step 12.17.2.2

Cancel the common factor.

$- \frac{2 \sqrt[3]{226}}{2 \cdot 2}$

Step 12.17.2.3

Rewrite the expression.

$- \frac{\sqrt[3]{226}}{2}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt[3]{226}}{2}$

Decimal Form:

$- 3.04559967 \dots$