Calculus Examples

$\frac{lim_{x \to 0} (2) \sin (6 x)}{5 \sin (4 x)}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to 0} \sin (6 x)}{5 \sin (4 x)}$

Step 1.2

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

$\frac{2 \sin (lim_{x \to 0} 6 x)}{5 \sin (4 x)}$

Step 1.3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 \sin (6 lim_{x \to 0} x)}{5 \sin (4 x)}$

Step 2

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

$\frac{2 \sin (6 \cdot 0)}{5 \sin (4 x)}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $6$ by $0$ .

$\frac{2 \sin (0)}{5 \sin (4 x)}$

Step 3.1.2

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

$\frac{2 \cdot 0}{5 \sin (4 x)}$

Step 3.2

Multiply $2$ by $0$ .

$\frac{0}{5 \sin (4 x)}$

Step 3.3

Cancel the common factor of $0$ and $5$ .

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Step 3.3.1

Factor $5$ out of $0$ .

$\frac{5 (0)}{5 \sin (4 x)}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $5$ out of $5 \sin (4 x)$ .

$\frac{5 (0)}{5 (\sin (4 x))}$

Step 3.3.2.2

Cancel the common factor.

$\frac{5 \cdot 0}{5 \sin (4 x)}$

Step 3.3.2.3

Rewrite the expression.

$\frac{0}{\sin (4 x)}$

Step 3.4

Separate fractions.

$\frac{0}{1} \cdot \frac{1}{\sin (4 x)}$

Step 3.5

Convert from $\frac{1}{\sin (4 x)}$ to $\csc (4 x)$ .

$\frac{0}{1} \csc (4 x)$

Step 3.6

Divide $0$ by $1$ .

$0 \csc (4 x)$

Step 3.7

Multiply $0$ by $\csc (4 x)$ .

$0$