Calculus Examples

$lim_{x \to π} \sin (x - \sin (x))$

Step 1

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

$\sin (lim_{x \to π} x - \sin (x))$

Step 2

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

$\sin (lim_{x \to π} x - lim_{x \to π} \sin (x))$

Step 3

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

$\sin (lim_{x \to π} x - \sin (lim_{x \to π} x))$

Step 4

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

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Step 4.1

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

$\sin (π - \sin (lim_{x \to π} x))$

Step 4.2

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

$\sin (π - \sin (π))$

Step 5

Simplify the answer.

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Step 5.1

Simplify each term.

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Step 5.1.1

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

$\sin (π - \sin (0))$

Step 5.1.2

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

$\sin (π - 0)$

Step 5.1.3

Multiply $- 1$ by $0$ .

$\sin (π + 0)$

Step 5.2

Add $π$ and $0$ .

$\sin (π)$

Step 5.3

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

$\sin (0)$

Step 5.4

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

$0$