Calculus Examples

$lim_{x \to 6} \frac{x^{2} + 4}{\sqrt{2 x + 5}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $6$ .

$\frac{lim_{x \to 6} x^{2} + 4}{lim_{x \to 6} \sqrt{2 x + 5}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$\frac{lim_{x \to 6} x^{2} + lim_{x \to 6} 4}{lim_{x \to 6} \sqrt{2 x + 5}}$

Step 3

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

$\frac{{(lim_{x \to 6} x)}^{2} + lim_{x \to 6} 4}{lim_{x \to 6} \sqrt{2 x + 5}}$

Step 4

Evaluate the limit of $4$ which is constant as $x$ approaches $6$ .

$\frac{{(lim_{x \to 6} x)}^{2} + 4}{lim_{x \to 6} \sqrt{2 x + 5}}$

Step 5

Move the limit under the radical sign.

$\frac{{(lim_{x \to 6} x)}^{2} + 4}{\sqrt{lim_{x \to 6} 2 x + 5}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$\frac{{(lim_{x \to 6} x)}^{2} + 4}{\sqrt{lim_{x \to 6} 2 x + lim_{x \to 6} 5}}$

Step 7

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 6} x)}^{2} + 4}{\sqrt{2 lim_{x \to 6} x + lim_{x \to 6} 5}}$

Step 8

Evaluate the limit of $5$ which is constant as $x$ approaches $6$ .

$\frac{{(lim_{x \to 6} x)}^{2} + 4}{\sqrt{2 lim_{x \to 6} x + 5}}$

Step 9

Evaluate the limits by plugging in $6$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{6^{2} + 4}{\sqrt{2 lim_{x \to 6} x + 5}}$

Step 9.2

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{6^{2} + 4}{\sqrt{2 \cdot 6 + 5}}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $6$ to the power of $2$ .

$\frac{36 + 4}{\sqrt{2 \cdot 6 + 5}}$

Step 10.1.2

Add $36$ and $4$ .

$\frac{40}{\sqrt{2 \cdot 6 + 5}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Multiply $2$ by $6$ .

$\frac{40}{\sqrt{12 + 5}}$

Step 10.2.2

Add $12$ and $5$ .

$\frac{40}{\sqrt{17}}$

Step 10.3

Multiply $\frac{40}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\frac{40}{\sqrt{17}} \cdot \frac{\sqrt{17}}{\sqrt{17}}$

Step 10.4

Combine and simplify the denominator.

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Step 10.4.1

Multiply $\frac{40}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\frac{40 \sqrt{17}}{\sqrt{17} \sqrt{17}}$

Step 10.4.2

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

$\frac{40 \sqrt{17}}{{\sqrt{17}}^{1} \sqrt{17}}$

Step 10.4.3

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

$\frac{40 \sqrt{17}}{{\sqrt{17}}^{1} {\sqrt{17}}^{1}}$

Step 10.4.4

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

$\frac{40 \sqrt{17}}{{\sqrt{17}}^{1 + 1}}$

Step 10.4.5

Add $1$ and $1$ .

$\frac{40 \sqrt{17}}{{\sqrt{17}}^{2}}$

Step 10.4.6

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

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Step 10.4.6.1

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

$\frac{40 \sqrt{17}}{{(17^{\frac{1}{2}})}^{2}}$

Step 10.4.6.2

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

$\frac{40 \sqrt{17}}{17^{\frac{1}{2} \cdot 2}}$

Step 10.4.6.3

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

$\frac{40 \sqrt{17}}{17^{\frac{2}{2}}}$

Step 10.4.6.4

Cancel the common factor of $2$ .

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Step 10.4.6.4.1

Cancel the common factor.

$\frac{40 \sqrt{17}}{17^{\frac{2}{2}}}$

Step 10.4.6.4.2

Rewrite the expression.

$\frac{40 \sqrt{17}}{17^{1}}$

Step 10.4.6.5

Evaluate the exponent.

$\frac{40 \sqrt{17}}{17}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{40 \sqrt{17}}{17}$

Decimal Form:

$9.70142500 \dots$