Calculus Examples

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

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 $x$ approaches $0$ .

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

Step 1.2

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

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

Step 1.3

Move the limit under the radical sign.

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Move the limit inside the trig function because cosine is continuous.

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

The exact value of $\cos (0)$ is $1$ .

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

Step 3.1.2

Add $1$ and $1$ .

$\frac{\sqrt{2} - \sqrt{2}}{\sin^{2} (x)}$

Step 3.1.3

Subtract $\sqrt{2}$ from $\sqrt{2}$ .

$\frac{0}{\sin^{2} (x)}$

Step 3.2

Multiply by $1$ .

$\frac{0}{\sin^{2} (x) \cdot 1}$

Step 3.3

Separate fractions.

$\frac{0}{1} \cdot \frac{1}{\sin^{2} (x)}$

Step 3.4

Convert from $\frac{1}{\sin^{2} (x)}$ to $\csc^{2} (x)$ .

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

Step 3.5

Divide $0$ by $1$ .

$0 \csc^{2} (x)$

Step 3.6

Multiply $0$ by $\csc^{2} (x)$ .

$0$