Calculus Examples

$lim_{x \to - 8} \frac{5 x}{\sqrt[4]{5 x^{4} + 3}}$

Step 1

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

$5 lim_{x \to - 8} \frac{x}{\sqrt[4]{5 x^{4} + 3}}$

Step 2

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

$5 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} \sqrt[4]{5 x^{4} + 3}}$

Step 3

Move the limit under the radical sign.

$5 \frac{lim_{x \to - 8} x}{\sqrt[4]{lim_{x \to - 8} 5 x^{4} + 3}}$

Step 4

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

$5 \frac{lim_{x \to - 8} x}{\sqrt[4]{lim_{x \to - 8} 5 x^{4} + lim_{x \to - 8} 3}}$

Step 5

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

$5 \frac{lim_{x \to - 8} x}{\sqrt[4]{5 lim_{x \to - 8} x^{4} + lim_{x \to - 8} 3}}$

Step 6

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

$5 \frac{lim_{x \to - 8} x}{\sqrt[4]{5 {(lim_{x \to - 8} x)}^{4} + lim_{x \to - 8} 3}}$

Step 7

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

$5 \frac{lim_{x \to - 8} x}{\sqrt[4]{5 {(lim_{x \to - 8} x)}^{4} + 3}}$

Step 8

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

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Step 8.1

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

$5 \frac{- 8}{\sqrt[4]{5 {(lim_{x \to - 8} x)}^{4} + 3}}$

Step 8.2

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

$5 \frac{- 8}{\sqrt[4]{5 {(- 8)}^{4} + 3}}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

Raise $- 8$ to the power of $4$ .

$5 \frac{- 8}{\sqrt[4]{5 \cdot 4096 + 3}}$

Step 9.1.2

Multiply $5$ by $4096$ .

$5 \frac{- 8}{\sqrt[4]{20480 + 3}}$

Step 9.1.3

Add $20480$ and $3$ .

$5 \frac{- 8}{\sqrt[4]{20483}}$

Step 9.2

Move the negative in front of the fraction.

$5 (- \frac{8}{\sqrt[4]{20483}})$

Step 9.3

Multiply $\frac{8}{\sqrt[4]{20483}}$ by $\frac{{\sqrt[4]{20483}}^{3}}{{\sqrt[4]{20483}}^{3}}$ .

$5 (- (\frac{8}{\sqrt[4]{20483}} \cdot \frac{{\sqrt[4]{20483}}^{3}}{{\sqrt[4]{20483}}^{3}}))$

Step 9.4

Combine and simplify the denominator.

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Step 9.4.1

Multiply $\frac{8}{\sqrt[4]{20483}}$ by $\frac{{\sqrt[4]{20483}}^{3}}{{\sqrt[4]{20483}}^{3}}$ .

$5 (- \frac{8 {\sqrt[4]{20483}}^{3}}{\sqrt[4]{20483} {\sqrt[4]{20483}}^{3}})$

Step 9.4.2

Raise $\sqrt[4]{20483}$ to the power of $1$ .

$5 (- \frac{8 {\sqrt[4]{20483}}^{3}}{{\sqrt[4]{20483}}^{1} {\sqrt[4]{20483}}^{3}})$

Step 9.4.3

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

$5 (- \frac{8 {\sqrt[4]{20483}}^{3}}{{\sqrt[4]{20483}}^{1 + 3}})$

Step 9.4.4

Add $1$ and $3$ .

$5 (- \frac{8 {\sqrt[4]{20483}}^{3}}{{\sqrt[4]{20483}}^{4}})$

Step 9.4.5

Rewrite ${\sqrt[4]{20483}}^{4}$ as $20483$ .

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Step 9.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{20483}$ as $20483^{\frac{1}{4}}$ .

$5 (- \frac{8 {\sqrt[4]{20483}}^{3}}{{(20483^{\frac{1}{4}})}^{4}})$

Step 9.4.5.2

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

$5 (- \frac{8 {\sqrt[4]{20483}}^{3}}{20483^{\frac{1}{4} \cdot 4}})$

Step 9.4.5.3

Combine $\frac{1}{4}$ and $4$ .

$5 (- \frac{8 {\sqrt[4]{20483}}^{3}}{20483^{\frac{4}{4}}})$

Step 9.4.5.4

Cancel the common factor of $4$ .

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Step 9.4.5.4.1

Cancel the common factor.

$5 (- \frac{8 {\sqrt[4]{20483}}^{3}}{20483^{\frac{4}{4}}})$

Step 9.4.5.4.2

Rewrite the expression.

$5 (- \frac{8 {\sqrt[4]{20483}}^{3}}{20483^{1}})$

Step 9.4.5.5

Evaluate the exponent.

$5 (- \frac{8 {\sqrt[4]{20483}}^{3}}{20483})$

Step 9.5

Simplify the numerator.

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Step 9.5.1

Rewrite ${\sqrt[4]{20483}}^{3}$ as $\sqrt[4]{20483^{3}}$ .

$5 (- \frac{8 \sqrt[4]{20483^{3}}}{20483})$

Step 9.5.2

Raise $20483$ to the power of $3$ .

$5 (- \frac{8 \sqrt[4]{8593710018587}}{20483})$

Step 9.6

Multiply $5 (- \frac{8 \sqrt[4]{8593710018587}}{20483})$ .

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Step 9.6.1

Multiply $- 1$ by $5$ .

$- 5 \frac{8 \sqrt[4]{8593710018587}}{20483}$

Step 9.6.2

Combine $- 5$ and $\frac{8 \sqrt[4]{8593710018587}}{20483}$ .

$\frac{- 5 (8 \sqrt[4]{8593710018587})}{20483}$

Step 9.6.3

Multiply $8$ by $- 5$ .

$\frac{- 40 \sqrt[4]{8593710018587}}{20483}$

Step 9.7

Move the negative in front of the fraction.

$- \frac{40 \sqrt[4]{8593710018587}}{20483}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{40 \sqrt[4]{8593710018587}}{20483}$

Decimal Form:

$- 3.34357908 \dots$