Calculus Examples

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

Step 1

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

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

Step 2

Move the limit under the radical sign.

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

Step 3

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

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

Step 4

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify the numerator.

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Step 7.1

Raising $0$ to any positive power yields $0$ .

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

Step 7.2

Add $0$ and $1$ .

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

Step 7.3

Any root of $1$ is $1$ .

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

Step 7.4

Add $1$ and $0$ .

$\frac{1}{x}$