Calculus Examples

$lim_{x \to 15} \frac{2}{\sqrt{x + 5}}$

Step 1

Evaluate the limit.

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Step 1.1

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

$2 lim_{x \to 15} \frac{1}{\sqrt{x + 5}}$

Step 1.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $15$ .

$2 \frac{lim_{x \to 15} 1}{lim_{x \to 15} \sqrt{x + 5}}$

Step 1.3

Evaluate the limit of $1$ which is constant as $x$ approaches $15$ .

$2 \frac{1}{lim_{x \to 15} \sqrt{x + 5}}$

Step 1.4

Move the limit under the radical sign.

$2 \frac{1}{\sqrt{lim_{x \to 15} x + 5}}$

Step 1.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $15$ .

$2 \frac{1}{\sqrt{lim_{x \to 15} x + lim_{x \to 15} 5}}$

Step 1.6

Evaluate the limit of $5$ which is constant as $x$ approaches $15$ .

$2 \frac{1}{\sqrt{lim_{x \to 15} x + 5}}$

Step 2

Evaluate the limit of $x$ by plugging in $15$ for $x$ .

$2 \frac{1}{\sqrt{15 + 5}}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

Add $15$ and $5$ .

$2 \frac{1}{\sqrt{20}}$

Step 3.1.2

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 3.1.2.1

Factor $4$ out of $20$ .

$2 \frac{1}{\sqrt{4 (5)}}$

Step 3.1.2.2

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

$2 \frac{1}{\sqrt{2^{2} \cdot 5}}$

Step 3.1.3

Pull terms out from under the radical.

$2 \frac{1}{2 \sqrt{5}}$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Factor $2$ out of $2 \sqrt{5}$ .

$2 \frac{1}{2 (\sqrt{5})}$

Step 3.2.2

Cancel the common factor.

$2 \frac{1}{2 \sqrt{5}}$

Step 3.2.3

Rewrite the expression.

$\frac{1}{\sqrt{5}}$

Step 3.3

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 3.4

Combine and simplify the denominator.

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Step 3.4.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 3.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 3.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 3.4.4

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

$\frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 3.4.5

Add $1$ and $1$ .

$\frac{\sqrt{5}}{{\sqrt{5}}^{2}}$

Step 3.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 3.4.6.2

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

$\frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 3.4.6.3

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

$\frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.4.6.4

Cancel the common factor of $2$ .

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Step 3.4.6.4.1

Cancel the common factor.

$\frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.4.6.4.2

Rewrite the expression.

$\frac{\sqrt{5}}{5^{1}}$

Step 3.4.6.5

Evaluate the exponent.

$\frac{\sqrt{5}}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{5}}{5}$

Decimal Form:

$0.44721359 \dots$