Calculus Examples

$lim_{x \to π} \sin (- 6 x + \sin (3 x))$

Step 1

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

$\sin (lim_{x \to π} - 6 x + \sin (3 x))$

Step 2

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

$\sin (- lim_{x \to π} 6 x + lim_{x \to π} \sin (3 x))$

Step 3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\sin (- (6 lim_{x \to π} x) + lim_{x \to π} \sin (3 x))$

Step 4

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

$\sin (- (6 lim_{x \to π} x) + \sin (lim_{x \to π} 3 x))$

Step 5

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

$\sin (- (6 lim_{x \to π} x) + \sin (3 lim_{x \to π} x))$

Step 6

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

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Step 6.1

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

$\sin (- (6 π) + \sin (3 lim_{x \to π} x))$

Step 6.2

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

$\sin (- (6 π) + \sin (3 π))$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $6$ by $- 1$ .

$\sin (- 6 π + \sin (3 π))$

Step 7.1.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\sin (- 6 π + \sin (π))$

Step 7.1.3

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

$\sin (- 6 π + \sin (0))$

Step 7.1.4

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

$\sin (- 6 π + 0)$

Step 7.2

Add $- 6 π$ and $0$ .

$\sin (- 6 π)$

Step 7.3

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\sin (0)$

Step 7.4

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

$0$