Calculus Examples

$\frac{lim_{x \to 0} \sqrt{1 + 3 x} - \sqrt{1 - 2 x}}{x}$

Step 1

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

$\frac{lim_{x \to 0} \sqrt{1 + 3 x} - lim_{x \to 0} \sqrt{1 - 2 x}}{x}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 0} 1 + 3 x} - lim_{x \to 0} \sqrt{1 - 2 x}}{x}$

Step 3

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

$\frac{\sqrt{lim_{x \to 0} 1 + lim_{x \to 0} 3 x} - lim_{x \to 0} \sqrt{1 - 2 x}}{x}$

Step 4

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

$\frac{\sqrt{1 + lim_{x \to 0} 3 x} - lim_{x \to 0} \sqrt{1 - 2 x}}{x}$

Step 5

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

$\frac{\sqrt{1 + 3 lim_{x \to 0} x} - lim_{x \to 0} \sqrt{1 - 2 x}}{x}$

Step 6

Move the limit under the radical sign.

$\frac{\sqrt{1 + 3 lim_{x \to 0} x} - \sqrt{lim_{x \to 0} 1 - 2 x}}{x}$

Step 7

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

$\frac{\sqrt{1 + 3 lim_{x \to 0} x} - \sqrt{lim_{x \to 0} 1 - lim_{x \to 0} 2 x}}{x}$

Step 8

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

$\frac{\sqrt{1 + 3 lim_{x \to 0} x} - \sqrt{1 - lim_{x \to 0} 2 x}}{x}$

Step 9

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

$\frac{\sqrt{1 + 3 lim_{x \to 0} x} - \sqrt{1 - 2 lim_{x \to 0} x}}{x}$

Step 10

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 10.1

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

$\frac{\sqrt{1 + 3 \cdot 0} - \sqrt{1 - 2 lim_{x \to 0} x}}{x}$

Step 10.2

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

$\frac{\sqrt{1 + 3 \cdot 0} - \sqrt{1 - 2 \cdot 0}}{x}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Multiply $3$ by $0$ .

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

Step 11.1.2

Add $1$ and $0$ .

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

Step 11.1.3

Any root of $1$ is $1$ .

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

Step 11.1.4

Multiply $- 2$ by $0$ .

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

Step 11.1.5

Add $1$ and $0$ .

$\frac{1 - \sqrt{1}}{x}$

Step 11.1.6

Any root of $1$ is $1$ .

$\frac{1 - 1 \cdot 1}{x}$

Step 11.1.7

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{x}$

Step 11.1.8

Subtract $1$ from $1$ .

$\frac{0}{x}$

Step 11.2

Divide $0$ by $x$ .

$0$