Calculus Examples

$\frac{lim_{x \to - 1} \sin (x^{2} - x - 2)}{x - 1}$

Step 1

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

$\frac{\sin (lim_{x \to - 1} x^{2} - x - 2)}{x - 1}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

$\frac{\sin (lim_{x \to - 1} x^{2} - lim_{x \to - 1} x - lim_{x \to - 1} 2)}{x - 1}$

Step 3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{\sin ({(lim_{x \to - 1} x)}^{2} - lim_{x \to - 1} x - lim_{x \to - 1} 2)}{x - 1}$

Step 4

Evaluate the limit of $2$ which is constant as $x$ approaches $- 1$ .

$\frac{\sin ({(lim_{x \to - 1} x)}^{2} - lim_{x \to - 1} x - 1 \cdot 2)}{x - 1}$

Step 5

Evaluate the limits by plugging in $- 1$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{\sin ({(- 1)}^{2} - lim_{x \to - 1} x - 1 \cdot 2)}{x - 1}$

Step 5.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{\sin ({(- 1)}^{2} + 1 - 1 \cdot 2)}{x - 1}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Simplify each term.

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Step 6.1.1.1

Raise $- 1$ to the power of $2$ .

$\frac{\sin (1 + 1 - 1 \cdot 2)}{x - 1}$

Step 6.1.1.2

Multiply $- 1$ by $2$ .

$\frac{\sin (1 + 1 - 2)}{x - 1}$

Step 6.1.2

Add $1$ and $1$ .

$\frac{\sin (2 - 2)}{x - 1}$

Step 6.1.3

Subtract $2$ from $2$ .

$\frac{\sin (0)}{x - 1}$

Step 6.1.4

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

$\frac{0}{x - 1}$

Step 6.2

Divide $0$ by $x - 1$ .

$0$