Calculus Examples

$lim_{h \to 0} \frac{2}{\sqrt{2 h + 1} + 1}$

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 $h$ .

$2 lim_{h \to 0} \frac{1}{\sqrt{2 h + 1} + 1}$

Step 1.2

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $0$ .

$2 \frac{lim_{h \to 0} 1}{lim_{h \to 0} \sqrt{2 h + 1} + 1}$

Step 1.3

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$2 \frac{1}{lim_{h \to 0} \sqrt{2 h + 1} + 1}$

Step 1.4

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$2 \frac{1}{lim_{h \to 0} \sqrt{2 h + 1} + lim_{h \to 0} 1}$

Step 1.5

Move the limit under the radical sign.

$2 \frac{1}{\sqrt{lim_{h \to 0} 2 h + 1} + lim_{h \to 0} 1}$

Step 1.6

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$2 \frac{1}{\sqrt{lim_{h \to 0} 2 h + lim_{h \to 0} 1} + lim_{h \to 0} 1}$

Step 1.7

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

$2 \frac{1}{\sqrt{2 lim_{h \to 0} h + lim_{h \to 0} 1} + lim_{h \to 0} 1}$

Step 1.8

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$2 \frac{1}{\sqrt{2 lim_{h \to 0} h + 1} + lim_{h \to 0} 1}$

Step 1.9

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$2 \frac{1}{\sqrt{2 lim_{h \to 0} h + 1} + 1}$

Step 2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$2 \frac{1}{\sqrt{2 \cdot 0 + 1} + 1}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

Multiply $2$ by $0$ .

$2 \frac{1}{\sqrt{0 + 1} + 1}$

Step 3.1.2

Add $0$ and $1$ .

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

Step 3.1.3

Any root of $1$ is $1$ .

$2 \frac{1}{1 + 1}$

Step 3.1.4

Add $1$ and $1$ .

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

$1$