Calculus Examples

$lim_{x \to 2} \sin (x^{2} - 4)$

Step 1

Evaluate the limit.

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Step 1.1

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

$\sin (lim_{x \to 2} x^{2} - 4)$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\sin (lim_{x \to 2} x^{2} - lim_{x \to 2} 4)$

Step 1.3

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

$\sin ({(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 4)$

Step 1.4

Evaluate the limit of $4$ which is constant as $x$ approaches $2$ .

$\sin ({(lim_{x \to 2} x)}^{2} - 1 \cdot 4)$

Step 2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\sin (2^{2} - 1 \cdot 4)$

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

$\sin (4 - 1 \cdot 4)$

Step 3.1.2

Multiply $- 1$ by $4$ .

$\sin (4 - 4)$

Step 3.2

Subtract $4$ from $4$ .

$\sin (0)$

Step 3.3

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

$0$