Calculus Examples

$lim_{x \to 8} \frac{- 8 x}{\sqrt{36 x^{2} + 8}}$

Step 1

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

$- 8 lim_{x \to 8} \frac{x}{\sqrt{36 x^{2} + 8}}$

Step 2

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

$- 8 \frac{lim_{x \to 8} x}{lim_{x \to 8} \sqrt{36 x^{2} + 8}}$

Step 3

Move the limit under the radical sign.

$- 8 \frac{lim_{x \to 8} x}{\sqrt{lim_{x \to 8} 36 x^{2} + 8}}$

Step 4

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

$- 8 \frac{lim_{x \to 8} x}{\sqrt{lim_{x \to 8} 36 x^{2} + lim_{x \to 8} 8}}$

Step 5

Move the term $36$ outside of the limit because it is constant with respect to $x$ .

$- 8 \frac{lim_{x \to 8} x}{\sqrt{36 lim_{x \to 8} x^{2} + lim_{x \to 8} 8}}$

Step 6

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

$- 8 \frac{lim_{x \to 8} x}{\sqrt{36 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 8}}$

Step 7

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

$- 8 \frac{lim_{x \to 8} x}{\sqrt{36 {(lim_{x \to 8} x)}^{2} + 8}}$

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 $x$ .

$- 8 \frac{8}{\sqrt{36 {(lim_{x \to 8} x)}^{2} + 8}}$

Step 8.2

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

$- 8 \frac{8}{\sqrt{36 \cdot 8^{2} + 8}}$

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 $2$ .

$- 8 \frac{8}{\sqrt{36 \cdot 64 + 8}}$

Step 9.1.2

Multiply $36$ by $64$ .

$- 8 \frac{8}{\sqrt{2304 + 8}}$

Step 9.1.3

Add $2304$ and $8$ .

$- 8 \frac{8}{\sqrt{2312}}$

Step 9.1.4

Rewrite $2312$ as $34^{2} \cdot 2$ .

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Step 9.1.4.1

Factor $1156$ out of $2312$ .

$- 8 \frac{8}{\sqrt{1156 (2)}}$

Step 9.1.4.2

Rewrite $1156$ as $34^{2}$ .

$- 8 \frac{8}{\sqrt{34^{2} \cdot 2}}$

Step 9.1.5

Pull terms out from under the radical.

$- 8 \frac{8}{34 \sqrt{2}}$

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

Factor $2$ out of $- 8$ .

$2 (- 4) \frac{8}{34 \sqrt{2}}$

Step 9.2.2

Factor $2$ out of $34 \sqrt{2}$ .

$2 (- 4) \frac{8}{2 (17 \sqrt{2})}$

Step 9.2.3

Cancel the common factor.

$2 \cdot - 4 \frac{8}{2 (17 \sqrt{2})}$

Step 9.2.4

Rewrite the expression.

$- 4 \frac{8}{17 \sqrt{2}}$

Step 9.3

Combine $- 4$ and $\frac{8}{17 \sqrt{2}}$ .

$\frac{- 4 \cdot 8}{17 \sqrt{2}}$

Step 9.4

Multiply $- 4$ by $8$ .

$\frac{- 32}{17 \sqrt{2}}$

Step 9.5

Move the negative in front of the fraction.

$- \frac{32}{17 \sqrt{2}}$

Step 9.6

Multiply $\frac{32}{17 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- (\frac{32}{17 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 9.7

Combine and simplify the denominator.

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Step 9.7.1

Multiply $\frac{32}{17 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- \frac{32 \sqrt{2}}{17 \sqrt{2} \sqrt{2}}$

Step 9.7.2

Move $\sqrt{2}$ .

$- \frac{32 \sqrt{2}}{17 (\sqrt{2} \sqrt{2})}$

Step 9.7.3

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

$- \frac{32 \sqrt{2}}{17 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 9.7.4

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

$- \frac{32 \sqrt{2}}{17 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 9.7.5

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

$- \frac{32 \sqrt{2}}{17 {\sqrt{2}}^{1 + 1}}$

Step 9.7.6

Add $1$ and $1$ .

$- \frac{32 \sqrt{2}}{17 {\sqrt{2}}^{2}}$

Step 9.7.7

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

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Step 9.7.7.1

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

$- \frac{32 \sqrt{2}}{17 {(2^{\frac{1}{2}})}^{2}}$

Step 9.7.7.2

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

$- \frac{32 \sqrt{2}}{17 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 9.7.7.3

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

$- \frac{32 \sqrt{2}}{17 \cdot 2^{\frac{2}{2}}}$

Step 9.7.7.4

Cancel the common factor of $2$ .

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Step 9.7.7.4.1

Cancel the common factor.

$- \frac{32 \sqrt{2}}{17 \cdot 2^{\frac{2}{2}}}$

Step 9.7.7.4.2

Rewrite the expression.

$- \frac{32 \sqrt{2}}{17 \cdot 2^{1}}$

Step 9.7.7.5

Evaluate the exponent.

$- \frac{32 \sqrt{2}}{17 \cdot 2}$

Step 9.8

Cancel the common factor of $32$ and $2$ .

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Step 9.8.1

Factor $2$ out of $32 \sqrt{2}$ .

$- \frac{2 (16 \sqrt{2})}{17 \cdot 2}$

Step 9.8.2

Cancel the common factors.

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Step 9.8.2.1

Factor $2$ out of $17 \cdot 2$ .

$- \frac{2 (16 \sqrt{2})}{2 \cdot 17}$

Step 9.8.2.2

Cancel the common factor.

$- \frac{2 (16 \sqrt{2})}{2 \cdot 17}$

Step 9.8.2.3

Rewrite the expression.

$- \frac{16 \sqrt{2}}{17}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{16 \sqrt{2}}{17}$

Decimal Form:

$- 1.33102452 \dots$