Calculus Examples

$lim_{x \to 5} \frac{\sin (x - 5)}{x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $5$ .

$\frac{lim_{x \to 5} \sin (x - 5)}{lim_{x \to 5} x}$

Step 2

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

$\frac{\sin (lim_{x \to 5} x - 5)}{lim_{x \to 5} x}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{\sin (lim_{x \to 5} x - lim_{x \to 5} 5)}{lim_{x \to 5} x}$

Step 4

Evaluate the limit of $5$ which is constant as $x$ approaches $5$ .

$\frac{\sin (lim_{x \to 5} x - 1 \cdot 5)}{lim_{x \to 5} x}$

Step 5

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

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Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{\sin (5 - 1 \cdot 5)}{lim_{x \to 5} x}$

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{\sin (5 - 1 \cdot 5)}{5}$

Step 6

Simplify the answer.

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Simplify the numerator.

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Multiply $- 1$ by $5$ .

$\frac{\sin (5 - 5)}{5}$

Subtract $5$ from $5$ .

$\frac{\sin (0)}{5}$

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

$\frac{0}{5}$

Divide $0$ by $5$ .

$0$