Calculus Examples

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

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

Multiply $- 1$ by $1$ .

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

Step 3.1.2

Subtract $1$ from $1$ .

$\frac{\sin (0)}{x^{2} + x - 2}$

Step 3.1.3

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

$\frac{0}{x^{2} + x - 2}$

Step 3.2

Factor $x^{2} + x - 2$ using the AC method.

Tap for more steps...

Step 3.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 3.2.2

Write the factored form using these integers.

$\frac{0}{(x - 1) (x + 2)}$

Step 3.3

Divide $0$ by $(x - 1) (x + 2)$ .

$0$