Calculus Examples

$lim_{x \to - 8} \frac{4 x - 1}{\sqrt{2 + 3 x^{2}}}$

Step 1

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

$\frac{lim_{x \to - 8} 4 x - 1}{lim_{x \to - 8} \sqrt{2 + 3 x^{2}}}$

Step 2

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

$\frac{lim_{x \to - 8} 4 x - lim_{x \to - 8} 1}{lim_{x \to - 8} \sqrt{2 + 3 x^{2}}}$

Step 3

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

$\frac{4 lim_{x \to - 8} x - lim_{x \to - 8} 1}{lim_{x \to - 8} \sqrt{2 + 3 x^{2}}}$

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $- 8$ .

$\frac{4 lim_{x \to - 8} x - 1 \cdot 1}{lim_{x \to - 8} \sqrt{2 + 3 x^{2}}}$

Step 5

Move the limit under the radical sign.

$\frac{4 lim_{x \to - 8} x - 1 \cdot 1}{\sqrt{lim_{x \to - 8} 2 + 3 x^{2}}}$

Step 6

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

$\frac{4 lim_{x \to - 8} x - 1 \cdot 1}{\sqrt{lim_{x \to - 8} 2 + lim_{x \to - 8} 3 x^{2}}}$

Step 7

Evaluate the limit of $2$ which is constant as $x$ approaches $- 8$ .

$\frac{4 lim_{x \to - 8} x - 1 \cdot 1}{\sqrt{2 + lim_{x \to - 8} 3 x^{2}}}$

Step 8

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

$\frac{4 lim_{x \to - 8} x - 1 \cdot 1}{\sqrt{2 + 3 lim_{x \to - 8} x^{2}}}$

Step 9

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Multiply $4$ by $- 8$ .

$\frac{- 32 - 1 \cdot 1}{\sqrt{2 + 3 {(- 8)}^{2}}}$

Step 11.1.2

Multiply $- 1$ by $1$ .

$\frac{- 32 - 1}{\sqrt{2 + 3 {(- 8)}^{2}}}$

Step 11.1.3

Subtract $1$ from $- 32$ .

$\frac{- 33}{\sqrt{2 + 3 {(- 8)}^{2}}}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Raise $- 8$ to the power of $2$ .

$\frac{- 33}{\sqrt{2 + 3 \cdot 64}}$

Step 11.2.2

Multiply $3$ by $64$ .

$\frac{- 33}{\sqrt{2 + 192}}$

Step 11.2.3

Add $2$ and $192$ .

$\frac{- 33}{\sqrt{194}}$

Step 11.3

Move the negative in front of the fraction.

$- \frac{33}{\sqrt{194}}$

Step 11.4

Multiply $\frac{33}{\sqrt{194}}$ by $\frac{\sqrt{194}}{\sqrt{194}}$ .

$- (\frac{33}{\sqrt{194}} \cdot \frac{\sqrt{194}}{\sqrt{194}})$

Step 11.5

Combine and simplify the denominator.

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Step 11.5.1

Multiply $\frac{33}{\sqrt{194}}$ by $\frac{\sqrt{194}}{\sqrt{194}}$ .

$- \frac{33 \sqrt{194}}{\sqrt{194} \sqrt{194}}$

Step 11.5.2

Raise $\sqrt{194}$ to the power of $1$ .

$- \frac{33 \sqrt{194}}{{\sqrt{194}}^{1} \sqrt{194}}$

Step 11.5.3

Raise $\sqrt{194}$ to the power of $1$ .

$- \frac{33 \sqrt{194}}{{\sqrt{194}}^{1} {\sqrt{194}}^{1}}$

Step 11.5.4

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

$- \frac{33 \sqrt{194}}{{\sqrt{194}}^{1 + 1}}$

Step 11.5.5

Add $1$ and $1$ .

$- \frac{33 \sqrt{194}}{{\sqrt{194}}^{2}}$

Step 11.5.6

Rewrite ${\sqrt{194}}^{2}$ as $194$ .

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Step 11.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{194}$ as $194^{\frac{1}{2}}$ .

$- \frac{33 \sqrt{194}}{{(194^{\frac{1}{2}})}^{2}}$

Step 11.5.6.2

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

$- \frac{33 \sqrt{194}}{194^{\frac{1}{2} \cdot 2}}$

Step 11.5.6.3

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

$- \frac{33 \sqrt{194}}{194^{\frac{2}{2}}}$

Step 11.5.6.4

Cancel the common factor of $2$ .

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Step 11.5.6.4.1

Cancel the common factor.

$- \frac{33 \sqrt{194}}{194^{\frac{2}{2}}}$

Step 11.5.6.4.2

Rewrite the expression.

$- \frac{33 \sqrt{194}}{194^{1}}$

Step 11.5.6.5

Evaluate the exponent.

$- \frac{33 \sqrt{194}}{194}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- \frac{33 \sqrt{194}}{194}$

Decimal Form:

$- 2.36926192 \dots$