Calculus Examples

$lim_{x \to 0} \frac{\sin (3 \cos (x))}{4 \sec (x)}$

Step 1

Move the term $\frac{1}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{4} lim_{x \to 0} \frac{\sin (3 \cos (x))}{\sec (x)}$

Step 2

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

$\frac{1}{4} \cdot \frac{lim_{x \to 0} \sin (3 \cos (x))}{lim_{x \to 0} \sec (x)}$

Step 3

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

$\frac{1}{4} \cdot \frac{\sin (lim_{x \to 0} 3 \cos (x))}{lim_{x \to 0} \sec (x)}$

Step 4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{4} \cdot \frac{\sin (3 lim_{x \to 0} \cos (x))}{lim_{x \to 0} \sec (x)}$

Step 5

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

$\frac{1}{4} \cdot \frac{\sin (3 \cos (lim_{x \to 0} x))}{lim_{x \to 0} \sec (x)}$

Step 6

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

$\frac{1}{4} \cdot \frac{\sin (3 \cos (lim_{x \to 0} x))}{\sec (lim_{x \to 0} x)}$

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 $x$ .

$\frac{1}{4} \cdot \frac{\sin (3 \cos (0))}{\sec (lim_{x \to 0} x)}$

Step 7.2

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

$\frac{1}{4} \cdot \frac{\sin (3 \cos (0))}{\sec (0)}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

$\frac{1}{4} \cdot \frac{\sin (3 \cdot 1)}{\sec (0)}$

Step 8.1.2

Multiply $3$ by $1$ .

$\frac{1}{4} \cdot \frac{\sin (3)}{\sec (0)}$

Step 8.1.3

Evaluate $\sin (3)$ .

$\frac{1}{4} \cdot \frac{0.05233595}{\sec (0)}$

Step 8.2

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

$\frac{1}{4} \cdot \frac{0.05233595}{1}$

Step 8.3

Divide $0.05233595$ by $1$ .

$\frac{1}{4} \cdot 0.05233595$

Step 8.4

Combine $\frac{1}{4}$ and $0.05233595$ .

$\frac{0.05233595}{4}$

Step 8.5

Divide $0.05233595$ by $4$ .

$0.01308398$