Calculus Examples

$lim_{x \to 8} \sin (\frac{π x}{2 - 3 x})$

Step 1

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

$\sin (lim_{x \to 8} \frac{π x}{2 - 3 x})$

Step 2

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

$\sin (π lim_{x \to 8} \frac{x}{2 - 3 x})$

Step 3

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

$\sin (π \frac{lim_{x \to 8} x}{lim_{x \to 8} 2 - 3 x})$

Step 4

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

$\sin (π \frac{lim_{x \to 8} x}{lim_{x \to 8} 2 - lim_{x \to 8} 3 x})$

Step 5

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\sin (π \frac{lim_{x \to 8} x}{2 - lim_{x \to 8} 3 x})$

Step 6

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

$\sin (π \frac{lim_{x \to 8} x}{2 - (3 lim_{x \to 8} x)})$

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 $x$ .

$\sin (π \frac{8}{2 - (3 lim_{x \to 8} x)})$

Step 7.2

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

$\sin (π \frac{8}{2 - (3 \cdot 8)})$

Step 8

Simplify the answer.

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Step 8.1

Cancel the common factor of $8$ and $2 - (3 \cdot 8)$ .

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Step 8.1.1

Reorder terms.

$\sin (π \frac{8}{- 1 \cdot 3 \cdot 8 + 2})$

Step 8.1.2

Factor $2$ out of $8$ .

$\sin (π \frac{2 \cdot 4}{- 1 \cdot 3 \cdot 8 + 2})$

Step 8.1.3

Cancel the common factors.

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Step 8.1.3.1

Factor $2$ out of $- 1 \cdot 3 \cdot 8$ .

$\sin (π \frac{2 \cdot 4}{2 (- 1 \cdot 3 \cdot 4) + 2})$

Step 8.1.3.2

Factor $2$ out of $2$ .

$\sin (π \frac{2 \cdot 4}{2 (- 1 \cdot 3 \cdot 4) + 2 (1)})$

Step 8.1.3.3

Factor $2$ out of $2 (- 1 \cdot 3 \cdot 4) + 2 (1)$ .

$\sin (π \frac{2 \cdot 4}{2 (- 1 \cdot 3 \cdot 4 + 1)})$

Step 8.1.3.4

Cancel the common factor.

$\sin (π \frac{2 \cdot 4}{2 (- 1 \cdot 3 \cdot 4 + 1)})$

Step 8.1.3.5

Rewrite the expression.

$\sin (π \frac{4}{- 1 \cdot 3 \cdot 4 + 1})$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $- 1 \cdot 3 \cdot 4$ .

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Step 8.2.1.1

Multiply $- 1$ by $3$ .

$\sin (π \frac{4}{- 3 \cdot 4 + 1})$

Step 8.2.1.2

Multiply $- 3$ by $4$ .

$\sin (π \frac{4}{- 12 + 1})$

Step 8.2.2

Add $- 12$ and $1$ .

$\sin (π \frac{4}{- 11})$

Step 8.3

Move the negative in front of the fraction.

$\sin (π (- \frac{4}{11}))$

Step 8.4

Combine $π$ and $\frac{4}{11}$ .

$\sin (- \frac{π \cdot 4}{11})$

Step 8.5

Move $4$ to the left of $π$ .

$\sin (- \frac{4 \cdot π}{11})$

Step 8.6

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

$\sin (\frac{18 π}{11})$

Step 8.7

Evaluate $\sin (\frac{18 π}{11})$ .

$- 0.90963199$