Calculus Examples

$\frac{lim_{x \to 0} \sin (x) \cos (4 x)}{x + x \cos (5 x)}$

Step 1

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

$\frac{lim_{x \to 0} \sin (x) \cdot lim_{x \to 0} \cos (4 x)}{x + x \cos (5 x)}$

Step 2

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

$\frac{\sin (lim_{x \to 0} x) \cdot lim_{x \to 0} \cos (4 x)}{x + x \cos (5 x)}$

Step 3

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

$\frac{\sin (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} 4 x)}{x + x \cos (5 x)}$

Step 4

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x)}{x + x \cos (5 x)}$

Step 5

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

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Step 5.1

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

$\frac{\sin (0) \cdot \cos (4 lim_{x \to 0} x)}{x + x \cos (5 x)}$

Step 5.2

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

$\frac{\sin (0) \cdot \cos (4 \cdot 0)}{x + x \cos (5 x)}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

The exact value of $\sin (0)$ is $0$ .

$\frac{0 \cos (4 \cdot 0)}{x + x \cos (5 x)}$

Step 6.1.2

Multiply $4$ by $0$ .

$\frac{0 \cos (0)}{x + x \cos (5 x)}$

Step 6.1.3

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

$\frac{0 \cdot 1}{x + x \cos (5 x)}$

Step 6.2

Factor $x$ out of $x + x \cos (5 x)$ .

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Step 6.2.1

Raise $x$ to the power of $1$ .

$\frac{0 \cdot 1}{x^{1} + x \cos (5 x)}$

Step 6.2.2

Factor $x$ out of $x^{1}$ .

$\frac{0 \cdot 1}{x \cdot 1 + x \cos (5 x)}$

Step 6.2.3

Factor $x$ out of $x \cos (5 x)$ .

$\frac{0 \cdot 1}{x \cdot 1 + x (\cos (5 x))}$

Step 6.2.4

Factor $x$ out of $x \cdot 1 + x (\cos (5 x))$ .

$\frac{0 \cdot 1}{x (1 + \cos (5 x))}$

Step 6.3

Multiply $0$ by $1$ .

$\frac{0}{x (1 + \cos (5 x))}$

Step 6.4

Divide $0$ by $x (1 + \cos (5 x))$ .

$0$