Calculus Examples

$lim_{x \to 8} \frac{x^{3} + 5}{\sqrt{x^{5}}}$

Step 1

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

$\frac{lim_{x \to 8} x^{3} + 5}{lim_{x \to 8} \sqrt{x^{5}}}$

Step 2

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

$\frac{lim_{x \to 8} x^{3} + lim_{x \to 8} 5}{lim_{x \to 8} \sqrt{x^{5}}}$

Step 3

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

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

Step 4

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

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

Step 5

Move the limit under the radical sign.

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

Step 6

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

$\frac{8^{3} + 5}{\sqrt{8^{5}}}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

$\frac{512 + 5}{\sqrt{8^{5}}}$

Step 8.1.2

Add $512$ and $5$ .

$\frac{517}{\sqrt{8^{5}}}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Raise $8$ to the power of $5$ .

$\frac{517}{\sqrt{32768}}$

Step 8.2.2

Rewrite $32768$ as $128^{2} \cdot 2$ .

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Step 8.2.2.1

Factor $16384$ out of $32768$ .

$\frac{517}{\sqrt{16384 (2)}}$

Step 8.2.2.2

Rewrite $16384$ as $128^{2}$ .

$\frac{517}{\sqrt{128^{2} \cdot 2}}$

Step 8.2.3

Pull terms out from under the radical.

$\frac{517}{128 \sqrt{2}}$

Step 8.3

Multiply $\frac{517}{128 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{517}{128 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 8.4

Combine and simplify the denominator.

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Step 8.4.1

Multiply $\frac{517}{128 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{517 \sqrt{2}}{128 \sqrt{2} \sqrt{2}}$

Step 8.4.2

Move $\sqrt{2}$ .

$\frac{517 \sqrt{2}}{128 (\sqrt{2} \sqrt{2})}$

Step 8.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{517 \sqrt{2}}{128 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 8.4.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{517 \sqrt{2}}{128 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 8.4.5

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

$\frac{517 \sqrt{2}}{128 {\sqrt{2}}^{1 + 1}}$

Step 8.4.6

Add $1$ and $1$ .

$\frac{517 \sqrt{2}}{128 {\sqrt{2}}^{2}}$

Step 8.4.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 8.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{517 \sqrt{2}}{128 {(2^{\frac{1}{2}})}^{2}}$

Step 8.4.7.2

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

$\frac{517 \sqrt{2}}{128 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 8.4.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{517 \sqrt{2}}{128 \cdot 2^{\frac{2}{2}}}$

Step 8.4.7.4

Cancel the common factor of $2$ .

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Step 8.4.7.4.1

Cancel the common factor.

$\frac{517 \sqrt{2}}{128 \cdot 2^{\frac{2}{2}}}$

Step 8.4.7.4.2

Rewrite the expression.

$\frac{517 \sqrt{2}}{128 \cdot 2^{1}}$

Step 8.4.7.5

Evaluate the exponent.

$\frac{517 \sqrt{2}}{128 \cdot 2}$

Step 8.5

Multiply $128$ by $2$ .

$\frac{517 \sqrt{2}}{256}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{517 \sqrt{2}}{256}$

Decimal Form:

$2.85604848 \dots$