Calculus Examples

$\frac{lim_{x \to 0} \sin (5 x)}{2 x}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{\sin (lim_{x \to 0} 5 x)}{2 x}$

Step 1.2

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (5 lim_{x \to 0} x)}{2 x}$

Step 2

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

$\frac{\sin (5 \cdot 0)}{2 x}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $5$ by $0$ .

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

Step 3.1.2

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

$\frac{0}{2 x}$

Step 3.2

Cancel the common factor of $0$ and $2$ .

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Step 3.2.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2 x}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $2$ out of $2 x$ .

$\frac{2 (0)}{2 (x)}$

Step 3.2.2.2

Cancel the common factor.

$\frac{2 \cdot 0}{2 x}$

Step 3.2.2.3

Rewrite the expression.

$\frac{0}{x}$

Step 3.3

Divide $0$ by $x$ .

$0$