Calculus Examples

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

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 1.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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 $x$ .

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

Step 6.2

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

$\sin (\cos (0)) \cdot \cos (0)$

Step 7

Simplify the answer.

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Step 7.1

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

$\sin (1) \cdot \cos (0)$

Step 7.2

Evaluate $\sin (1)$ .

$0.0174524 \cdot \cos (0)$

Step 7.3

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

$0.0174524 \cdot 1$

Step 7.4

Multiply $0.0174524$ by $1$ .

$0.0174524$