Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

$\frac{\cot (\frac{π}{2}) - \cot (\frac{π}{2})}{\frac{π}{2} - x}$

Step 3

Simplify the answer.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Combine.

$\frac{2 (\cot (\frac{π}{2}) - \cot (\frac{π}{2}))}{2 (\frac{π}{2} - x)}$

Step 3.2

Apply the distributive property.

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

Step 3.3

Cancel the common factor of $2$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

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

$\frac{2 \cdot 0 + 2 (- \cot (\frac{π}{2}))}{π + 2 (- x)}$

Step 3.4.2

Multiply $2$ by $0$ .

$\frac{0 + 2 (- \cot (\frac{π}{2}))}{π + 2 (- x)}$

Step 3.4.3

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

$\frac{0 + 2 (- 0)}{π + 2 (- x)}$

Step 3.4.4

Multiply $2 (- 0)$ .

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Step 3.4.4.1

Multiply $- 1$ by $0$ .

$\frac{0 + 2 \cdot 0}{π + 2 (- x)}$

Step 3.4.4.2

Multiply $2$ by $0$ .

$\frac{0 + 0}{π + 2 (- x)}$

Step 3.4.5

Add $0$ and $0$ .

$\frac{0}{π + 2 (- x)}$

Step 3.5

Multiply $- 1$ by $2$ .

$\frac{0}{π - 2 x}$

Step 3.6

Divide $0$ by $π - 2 x$ .

$0$