Calculus Examples

$\frac{lim_{x \to 14} \sin (x - 14)}{x^{2} - 196}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{\sin (lim_{x \to 14} x - 14)}{x^{2} - 196}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $14$ .

$\frac{\sin (lim_{x \to 14} x - lim_{x \to 14} 14)}{x^{2} - 196}$

Step 1.3

Evaluate the limit of $14$ which is constant as $x$ approaches $14$ .

$\frac{\sin (lim_{x \to 14} x - 1 \cdot 14)}{x^{2} - 196}$

Step 2

Evaluate the limit of $x$ by plugging in $14$ for $x$ .

$\frac{\sin (14 - 1 \cdot 14)}{x^{2} - 196}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $- 1$ by $14$ .

$\frac{\sin (14 - 14)}{x^{2} - 196}$

Step 3.1.2

Subtract $14$ from $14$ .

$\frac{\sin (0)}{x^{2} - 196}$

Step 3.1.3

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

$\frac{0}{x^{2} - 196}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite $196$ as $14^{2}$ .

$\frac{0}{x^{2} - 14^{2}}$

Step 3.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 14$ .

$\frac{0}{(x + 14) (x - 14)}$

Step 3.3

Divide $0$ by $(x + 14) (x - 14)$ .

$0$