Calculus Examples

$\frac{lim_{x \to 0} \sin (x^{2})}{6 x}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{\sin (lim_{x \to 0} x^{2})}{6 x}$

Step 1.2

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

$\frac{\sin ({(lim_{x \to 0} x)}^{2})}{6 x}$

Step 2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sin (0^{2})}{6 x}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{\sin (0)}{6 x}$

Step 3.1.2

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

$\frac{0}{6 x}$

Step 3.2

Cancel the common factor of $0$ and $6$ .

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Step 3.2.1

Factor $6$ out of $0$ .

$\frac{6 (0)}{6 x}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $6$ out of $6 x$ .

$\frac{6 (0)}{6 (x)}$

Step 3.2.2.2

Cancel the common factor.

$\frac{6 \cdot 0}{6 x}$

Step 3.2.2.3

Rewrite the expression.

$\frac{0}{x}$

Step 3.3

Divide $0$ by $x$ .

$0$