Precalculus Examples

$lim_{x \to \frac{π}{4}} \frac{1 - \cos (2 x)}{3 x \sin (2 x)}$

Step 1

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to \frac{π}{4}} \frac{1 - \cos (2 x)}{x \sin (2 x)}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{π}{4}$ .

$\frac{1}{3} \cdot \frac{lim_{x \to \frac{π}{4}} 1 - \cos (2 x)}{lim_{x \to \frac{π}{4}} x \sin (2 x)}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{4}$ .

$\frac{1}{3} \cdot \frac{lim_{x \to \frac{π}{4}} 1 - lim_{x \to \frac{π}{4}} \cos (2 x)}{lim_{x \to \frac{π}{4}} x \sin (2 x)}$

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{π}{4}$ .

$\frac{1}{3} \cdot \frac{1 - lim_{x \to \frac{π}{4}} \cos (2 x)}{lim_{x \to \frac{π}{4}} x \sin (2 x)}$

Step 5

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

$\frac{1}{3} \cdot \frac{1 - \cos (lim_{x \to \frac{π}{4}} 2 x)}{lim_{x \to \frac{π}{4}} x \sin (2 x)}$

Step 6

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

$\frac{1}{3} \cdot \frac{1 - \cos (2 lim_{x \to \frac{π}{4}} x)}{lim_{x \to \frac{π}{4}} x \sin (2 x)}$

Step 7

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $\frac{π}{4}$ .

$\frac{1}{3} \cdot \frac{1 - \cos (2 lim_{x \to \frac{π}{4}} x)}{lim_{x \to \frac{π}{4}} x \cdot lim_{x \to \frac{π}{4}} \sin (2 x)}$

Step 8

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

$\frac{1}{3} \cdot \frac{1 - \cos (2 lim_{x \to \frac{π}{4}} x)}{lim_{x \to \frac{π}{4}} x \cdot \sin (lim_{x \to \frac{π}{4}} 2 x)}$

Step 9

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

$\frac{1}{3} \cdot \frac{1 - \cos (2 lim_{x \to \frac{π}{4}} x)}{lim_{x \to \frac{π}{4}} x \cdot \sin (2 lim_{x \to \frac{π}{4}} x)}$

Step 10

Evaluate the limits by plugging in $\frac{π}{4}$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{1}{3} \cdot \frac{1 - \cos (2 \frac{π}{4})}{lim_{x \to \frac{π}{4}} x \cdot \sin (2 lim_{x \to \frac{π}{4}} x)}$

Step 10.2

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{1}{3} \cdot \frac{1 - \cos (2 \frac{π}{4})}{\frac{π}{4} \cdot \sin (2 lim_{x \to \frac{π}{4}} x)}$

Step 10.3

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{1}{3} \cdot \frac{1 - \cos (2 \frac{π}{4})}{\frac{π}{4} \cdot \sin (2 \frac{π}{4})}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Cancel the common factor of $2$ .

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Step 11.1.1.1

Factor $2$ out of $4$ .

$\frac{1}{3} \cdot \frac{1 - \cos (2 \frac{π}{2 (2)})}{\frac{π}{4} \cdot \sin (2 \frac{π}{4})}$

Step 11.1.1.2

Cancel the common factor.

$\frac{1}{3} \cdot \frac{1 - \cos (2 \frac{π}{2 \cdot 2})}{\frac{π}{4} \cdot \sin (2 \frac{π}{4})}$

Step 11.1.1.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{1 - \cos (\frac{π}{2})}{\frac{π}{4} \cdot \sin (2 \frac{π}{4})}$

Step 11.1.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{1}{3} \cdot \frac{1 - 0}{\frac{π}{4} \cdot \sin (2 \frac{π}{4})}$

Step 11.1.3

Multiply $- 1$ by $0$ .

$\frac{1}{3} \cdot \frac{1 + 0}{\frac{π}{4} \cdot \sin (2 \frac{π}{4})}$

Step 11.1.4

Add $1$ and $0$ .

$\frac{1}{3} \cdot \frac{1}{\frac{π}{4} \cdot \sin (2 \frac{π}{4})}$

Step 11.2

Combine $\frac{π}{4}$ and $\sin (2 \frac{π}{4})$ .

$\frac{1}{3} \cdot \frac{1}{\frac{π \sin (2 \frac{π}{4})}{4}}$

Step 11.3

Simplify the numerator.

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Step 11.3.1

Cancel the common factor of $2$ .

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Step 11.3.1.1

Factor $2$ out of $4$ .

$\frac{1}{3} \cdot \frac{1}{\frac{π \sin (2 \frac{π}{2 (2)})}{4}}$

Step 11.3.1.2

Cancel the common factor.

$\frac{1}{3} \cdot \frac{1}{\frac{π \sin (2 \frac{π}{2 \cdot 2})}{4}}$

Step 11.3.1.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{1}{\frac{π \sin (\frac{π}{2})}{4}}$

Step 11.3.2

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{1}{3} \cdot \frac{1}{\frac{π \cdot 1}{4}}$

Step 11.4

Multiply $π$ by $1$ .

$\frac{1}{3} \cdot \frac{1}{\frac{π}{4}}$

Step 11.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{3} (1 \frac{4}{π})$

Step 11.6

Multiply $\frac{4}{π}$ by $1$ .

$\frac{1}{3} \cdot \frac{4}{π}$

Step 11.7

Multiply $\frac{1}{3}$ by $\frac{4}{π}$ .

$\frac{4}{3 π}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{3 π}$

Decimal Form:

$0.42441318 \dots$