Calculus Examples

$lim_{x \to - 8} \sin (\frac{π x}{3 - 6 x})$

Step 1

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

$\sin (lim_{x \to - 8} \frac{π x}{3 - 6 x})$

Step 2

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

$\sin (π lim_{x \to - 8} \frac{x}{3 - 6 x})$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 8$ .

$\sin (π \frac{lim_{x \to - 8} x}{lim_{x \to - 8} 3 - 6 x})$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$\sin (π \frac{lim_{x \to - 8} x}{lim_{x \to - 8} 3 - lim_{x \to - 8} 6 x})$

Step 5

Evaluate the limit of $3$ which is constant as $x$ approaches $- 8$ .

$\sin (π \frac{lim_{x \to - 8} x}{3 - lim_{x \to - 8} 6 x})$

Step 6

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

$\sin (π \frac{lim_{x \to - 8} x}{3 - (6 lim_{x \to - 8} x)})$

Step 7

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$\sin (π \frac{- 8}{3 - (6 lim_{x \to - 8} x)})$

Step 7.2

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$\sin (π \frac{- 8}{3 - (6 \cdot - 8)})$

Step 8

Simplify the answer.

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Step 8.1

Simplify the denominator.

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Step 8.1.1

Multiply $- 1 \cdot 6 \cdot - 8$ .

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Step 8.1.1.1

Multiply $- 1$ by $6$ .

$\sin (π \frac{- 8}{3 - 6 \cdot - 8})$

Step 8.1.1.2

Multiply $- 6$ by $- 8$ .

$\sin (π \frac{- 8}{3 + 48})$

Step 8.1.2

Add $3$ and $48$ .

$\sin (π \frac{- 8}{51})$

Step 8.2

Move the negative in front of the fraction.

$\sin (π (- \frac{8}{51}))$

Step 8.3

Combine $π$ and $\frac{8}{51}$ .

$\sin (- \frac{π \cdot 8}{51})$

Step 8.4

Move $8$ to the left of $π$ .

$\sin (- \frac{8 \cdot π}{51})$

Step 8.5

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

$\sin (\frac{94 π}{51})$

Step 8.6

Evaluate $\sin (\frac{94 π}{51})$ .

$- 0.47309355$