Calculus Examples

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

Step 1

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

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

Step 2

Move the limit under the radical sign.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Add $1$ and $0$ .

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

Step 8.1.2

Any root of $1$ is $1$ .

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

Step 8.1.3

Multiply $- 1$ by $1$ .

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

Step 8.1.4

Multiply $- \frac{1}{2} \cdot 0$ .

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Step 8.1.4.1

Multiply $0$ by $- 1$ .

$\frac{1 - 1 + 0 (\frac{1}{2})}{x^{2}}$

Step 8.1.4.2

Multiply $0$ by $\frac{1}{2}$ .

$\frac{1 - 1 + 0}{x^{2}}$

Step 8.1.5

Add $1 - 1$ and $0$ .

$\frac{1 - 1}{x^{2}}$

Step 8.1.6

Subtract $1$ from $1$ .

$\frac{0}{x^{2}}$

Step 8.2

Divide $0$ by $x^{2}$ .

$0$