Calculus Examples

$lim_{x \to π} 4 \sin (x + \sin (x))$

Step 1

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

$4 lim_{x \to π} \sin (x + \sin (x))$

Step 2

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

$4 \sin (lim_{x \to π} x + \sin (x))$

Step 3

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

$4 \sin (lim_{x \to π} x + lim_{x \to π} \sin (x))$

Step 4

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

$4 \sin (lim_{x \to π} x + \sin (lim_{x \to π} x))$

Step 5

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

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Step 5.1

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

$4 \sin (π + \sin (lim_{x \to π} x))$

Step 5.2

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

$4 \sin (π + \sin (π))$

Step 6

Simplify the answer.

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Step 6.1

Simplify each term.

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Step 6.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$4 \sin (π + \sin (0))$

Step 6.1.2

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

$4 \sin (π + 0)$

Step 6.2

Add $π$ and $0$ .

$4 \sin (π)$

Step 6.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$4 \sin (0)$

Step 6.4

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

$4 \cdot 0$

Step 6.5

Multiply $4$ by $0$ .

$0$