Calculus Examples

$lim_{x \to - 8} \frac{3 x + 5}{\sqrt{x^{2} + 4}}$

Step 1

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

$\frac{lim_{x \to - 8} 3 x + 5}{lim_{x \to - 8} \sqrt{x^{2} + 4}}$

Step 2

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

$\frac{lim_{x \to - 8} 3 x + lim_{x \to - 8} 5}{lim_{x \to - 8} \sqrt{x^{2} + 4}}$

Step 3

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

$\frac{3 lim_{x \to - 8} x + lim_{x \to - 8} 5}{lim_{x \to - 8} \sqrt{x^{2} + 4}}$

Step 4

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

$\frac{3 lim_{x \to - 8} x + 5}{lim_{x \to - 8} \sqrt{x^{2} + 4}}$

Step 5

Move the limit under the radical sign.

$\frac{3 lim_{x \to - 8} x + 5}{\sqrt{lim_{x \to - 8} x^{2} + 4}}$

Step 6

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

$\frac{3 lim_{x \to - 8} x + 5}{\sqrt{lim_{x \to - 8} x^{2} + lim_{x \to - 8} 4}}$

Step 7

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

$\frac{3 lim_{x \to - 8} x + 5}{\sqrt{{(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} 4}}$

Step 8

Evaluate the limit of $4$ which is constant as $x$ approaches $- 8$ .

$\frac{3 lim_{x \to - 8} x + 5}{\sqrt{{(lim_{x \to - 8} x)}^{2} + 4}}$

Step 9

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

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Step 9.1

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

$\frac{3 \cdot - 8 + 5}{\sqrt{{(lim_{x \to - 8} x)}^{2} + 4}}$

Step 9.2

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

$\frac{3 \cdot - 8 + 5}{\sqrt{{(- 8)}^{2} + 4}}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Multiply $3$ by $- 8$ .

$\frac{- 24 + 5}{\sqrt{{(- 8)}^{2} + 4}}$

Step 10.1.2

Add $- 24$ and $5$ .

$\frac{- 19}{\sqrt{{(- 8)}^{2} + 4}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

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

$\frac{- 19}{\sqrt{64 + 4}}$

Step 10.2.2

Add $64$ and $4$ .

$\frac{- 19}{\sqrt{68}}$

Step 10.2.3

Rewrite $68$ as $2^{2} \cdot 17$ .

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Step 10.2.3.1

Factor $4$ out of $68$ .

$\frac{- 19}{\sqrt{4 (17)}}$

Step 10.2.3.2

Rewrite $4$ as $2^{2}$ .

$\frac{- 19}{\sqrt{2^{2} \cdot 17}}$

Step 10.2.4

Pull terms out from under the radical.

$\frac{- 19}{2 \sqrt{17}}$

Step 10.3

Move the negative in front of the fraction.

$- \frac{19}{2 \sqrt{17}}$

Step 10.4

Multiply $\frac{19}{2 \sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$- (\frac{19}{2 \sqrt{17}} \cdot \frac{\sqrt{17}}{\sqrt{17}})$

Step 10.5

Combine and simplify the denominator.

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Step 10.5.1

Multiply $\frac{19}{2 \sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$- \frac{19 \sqrt{17}}{2 \sqrt{17} \sqrt{17}}$

Step 10.5.2

Move $\sqrt{17}$ .

$- \frac{19 \sqrt{17}}{2 (\sqrt{17} \sqrt{17})}$

Step 10.5.3

Raise $\sqrt{17}$ to the power of $1$ .

$- \frac{19 \sqrt{17}}{2 ({\sqrt{17}}^{1} \sqrt{17})}$

Step 10.5.4

Raise $\sqrt{17}$ to the power of $1$ .

$- \frac{19 \sqrt{17}}{2 ({\sqrt{17}}^{1} {\sqrt{17}}^{1})}$

Step 10.5.5

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

$- \frac{19 \sqrt{17}}{2 {\sqrt{17}}^{1 + 1}}$

Step 10.5.6

Add $1$ and $1$ .

$- \frac{19 \sqrt{17}}{2 {\sqrt{17}}^{2}}$

Step 10.5.7

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 10.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$- \frac{19 \sqrt{17}}{2 {(17^{\frac{1}{2}})}^{2}}$

Step 10.5.7.2

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

$- \frac{19 \sqrt{17}}{2 \cdot 17^{\frac{1}{2} \cdot 2}}$

Step 10.5.7.3

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

$- \frac{19 \sqrt{17}}{2 \cdot 17^{\frac{2}{2}}}$

Step 10.5.7.4

Cancel the common factor of $2$ .

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Step 10.5.7.4.1

Cancel the common factor.

$- \frac{19 \sqrt{17}}{2 \cdot 17^{\frac{2}{2}}}$

Step 10.5.7.4.2

Rewrite the expression.

$- \frac{19 \sqrt{17}}{2 \cdot 17^{1}}$

Step 10.5.7.5

Evaluate the exponent.

$- \frac{19 \sqrt{17}}{2 \cdot 17}$

Step 10.6

Multiply $2$ by $17$ .

$- \frac{19 \sqrt{17}}{34}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{19 \sqrt{17}}{34}$

Decimal Form:

$- 2.30408843 \dots$