Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

Move the limit under the radical sign.

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Evaluate the limit of $\sqrt{1 + 2 x}$ which is constant as $h$ approaches $0$ .

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

Step 2

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

$\frac{\sqrt{1 + 2 x \cdot 0} - \sqrt{1 + 2 x}}{h}$

Step 3

Simplify the numerator.

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Step 3.1

Multiply $2 x \cdot 0$ .

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Step 3.1.1

Multiply $0$ by $2$ .

$\frac{\sqrt{1 + 0 x} - \sqrt{1 + 2 x}}{h}$

Step 3.1.2

Multiply $0$ by $x$ .

$\frac{\sqrt{1 + 0} - \sqrt{1 + 2 x}}{h}$

Step 3.2

Add $1$ and $0$ .

$\frac{\sqrt{1} - \sqrt{1 + 2 x}}{h}$

Step 3.3

Any root of $1$ is $1$ .

$\frac{1 - \sqrt{1 + 2 x}}{h}$