Calculus Examples

$\frac{lim_{x \to \frac{π}{2}} 2 x \sin (x) - π}{x - (\frac{π}{2})}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{2}$ .

$\frac{lim_{x \to \frac{π}{2}} 2 x \sin (x) - lim_{x \to \frac{π}{2}} π}{x - (\frac{π}{2})}$

Step 2

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

$\frac{2 lim_{x \to \frac{π}{2}} x \sin (x) - lim_{x \to \frac{π}{2}} π}{x - (\frac{π}{2})}$

Step 3

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $\frac{π}{2}$ .

$\frac{2 lim_{x \to \frac{π}{2}} x \cdot lim_{x \to \frac{π}{2}} \sin (x) - lim_{x \to \frac{π}{2}} π}{x - (\frac{π}{2})}$

Step 4

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

$\frac{2 lim_{x \to \frac{π}{2}} x \cdot \sin (lim_{x \to \frac{π}{2}} x) - lim_{x \to \frac{π}{2}} π}{x - (\frac{π}{2})}$

Step 5

Evaluate the limit of $π$ which is constant as $x$ approaches $\frac{π}{2}$ .

$\frac{2 lim_{x \to \frac{π}{2}} x \cdot \sin (lim_{x \to \frac{π}{2}} x) - π}{x - (\frac{π}{2})}$

Step 6

Evaluate the limits by plugging in $\frac{π}{2}$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$\frac{2 \frac{π}{2} \cdot \sin (lim_{x \to \frac{π}{2}} x) - π}{x - (\frac{π}{2})}$

Step 6.2

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$\frac{2 \frac{π}{2} \cdot \sin (\frac{π}{2}) - π}{x - (\frac{π}{2})}$

Step 7

Simplify the answer.

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Step 7.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 7.1.1

Multiply $\frac{2 \frac{π}{2} \cdot \sin (\frac{π}{2}) - π}{x - \frac{π}{2}}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{2 \frac{π}{2} \cdot \sin (\frac{π}{2}) - π}{x - \frac{π}{2}}$

Step 7.1.2

Combine.

$\frac{2 (2 \frac{π}{2} \cdot \sin (\frac{π}{2}) - π)}{2 (x - \frac{π}{2})}$

Step 7.2

Apply the distributive property.

$\frac{2 (2 \frac{π}{2} \cdot \sin (\frac{π}{2})) + 2 (- π)}{2 x + 2 (- \frac{π}{2})}$

Step 7.3

Cancel the common factor of $2$ .

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Step 7.3.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$\frac{2 (2 \frac{π}{2} \cdot \sin (\frac{π}{2})) + 2 (- π)}{2 x + 2 \frac{- π}{2}}$

Step 7.3.2

Cancel the common factor.

$\frac{2 (2 \frac{π}{2} \cdot \sin (\frac{π}{2})) + 2 (- π)}{2 x + 2 \frac{- π}{2}}$

Step 7.3.3

Rewrite the expression.

$\frac{2 (2 \frac{π}{2} \cdot \sin (\frac{π}{2})) + 2 (- π)}{2 x - π}$

Step 7.4

Simplify the numerator.

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Step 7.4.1

Multiply $2$ by $2$ .

$\frac{4 \frac{π}{2} \cdot \sin (\frac{π}{2}) + 2 (- π)}{2 x - π}$

Step 7.4.2

Cancel the common factor of $2$ .

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Step 7.4.2.1

Factor $2$ out of $4$ .

$\frac{2 (2) \frac{π}{2} \cdot \sin (\frac{π}{2}) + 2 (- π)}{2 x - π}$

Step 7.4.2.2

Cancel the common factor.

$\frac{2 \cdot 2 \frac{π}{2} \cdot \sin (\frac{π}{2}) + 2 (- π)}{2 x - π}$

Step 7.4.2.3

Rewrite the expression.

$\frac{2 π \cdot \sin (\frac{π}{2}) + 2 (- π)}{2 x - π}$

Step 7.4.3

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

$\frac{2 π \cdot 1 + 2 (- π)}{2 x - π}$

Step 7.4.4

Multiply $2$ by $1$ .

$\frac{2 π + 2 (- π)}{2 x - π}$

Step 7.4.5

Multiply $- 1$ by $2$ .

$\frac{2 π - 2 π}{2 x - π}$

Step 7.4.6

Subtract $2 π$ from $2 π$ .

$\frac{0}{2 x - π}$

Step 7.5

Divide $0$ by $2 x - π$ .

$0$