Calculus Examples

$\frac{lim_{x \to 0} x + x \cos (x)}{\sin (x) \cos (x)}$

Step 1

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

$\frac{lim_{x \to 0} x + lim_{x \to 0} x \cos (x)}{\sin (x) \cos (x)}$

Step 2

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

$\frac{lim_{x \to 0} x + lim_{x \to 0} x \cdot lim_{x \to 0} \cos (x)}{\sin (x) \cos (x)}$

Step 3

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

$\frac{lim_{x \to 0} x + lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x)}{\sin (x) \cos (x)}$

Step 4

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

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Step 4.1

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

$\frac{0 + lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x)}{\sin (x) \cos (x)}$

Step 4.2

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

$\frac{0 + 0 \cdot \cos (lim_{x \to 0} x)}{\sin (x) \cos (x)}$

Step 4.3

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

$\frac{0 + 0 \cdot \cos (0)}{\sin (x) \cos (x)}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$\frac{0 + 0 \cdot 1}{\sin (x) \cos (x)}$

Step 5.1.2

Multiply $0$ by $1$ .

$\frac{0 + 0}{\sin (x) \cos (x)}$

Step 5.1.3

Add $0$ and $0$ .

$\frac{0}{\sin (x) \cos (x)}$

Step 5.2

Separate fractions.

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

Step 5.3

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

$\frac{0}{\cos (x)} \csc (x)$

Step 5.4

Separate fractions.

$\frac{0}{1} \cdot \frac{1}{\cos (x)} \csc (x)$

Step 5.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{0}{1} \sec (x) \csc (x)$

Step 5.6

Divide $0$ by $1$ .

$0 \sec (x) \csc (x)$

Step 5.7

Multiply $0$ by $\sec (x)$ .

$0 \csc (x)$

Step 5.8

Multiply $0$ by $\csc (x)$ .

$0$