Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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 $x$ .

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

Step 4.2

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

$\cos (0 + \sin (0))$

Step 5

Simplify the answer.

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Step 5.1

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

$\cos (0 + 0)$

Step 5.2

Add $0$ and $0$ .

$\cos (0)$

Step 5.3

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

$1$