Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.2

Combine $x$ and $\frac{2}{2}$ .

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

Step 1.3

Combine the numerators over the common denominator.

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

Step 2

Evaluate the limit.

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Step 2.1

Simplify the limit argument.

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Step 2.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.2

Combine $\cos (x)$ and $\frac{2}{x \cdot 2 - π}$ .

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

Step 2.2

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

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

Step 3

Apply L'Hospital's rule.

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Step 3.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 3.1.1

Take the limit of the numerator and the limit of the denominator.

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

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

$2 \frac{0}{lim_{x \to \frac{π}{2}} x \cdot 2 - π}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Evaluate the limit.

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Step 3.1.3.1.1

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

$2 \frac{0}{lim_{x \to \frac{π}{2}} x \cdot 2 - lim_{x \to \frac{π}{2}} π}$

Step 3.1.3.1.2

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

$2 \frac{0}{2 lim_{x \to \frac{π}{2}} x - lim_{x \to \frac{π}{2}} π}$

Step 3.1.3.1.3

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

$2 \frac{0}{2 lim_{x \to \frac{π}{2}} x - π}$

Step 3.1.3.2

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

$2 \frac{0}{2 \frac{π}{2} - π}$

Step 3.1.3.3

Simplify the answer.

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Step 3.1.3.3.1

Cancel the common factor of $2$ .

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Step 3.1.3.3.1.1

Cancel the common factor.

$2 \frac{0}{2 \frac{π}{2} - π}$

Step 3.1.3.3.1.2

Rewrite the expression.

$2 \frac{0}{π - π}$

Step 3.1.3.3.2

Subtract $π$ from $π$ .

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

Step 3.1.3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

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

Step 3.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

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

Step 3.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

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

Step 3.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to \frac{π}{2}} \frac{\cos (x)}{x \cdot 2 - π} = lim_{x \to \frac{π}{2}} \frac{\frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x \cdot 2 - π]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$2 lim_{x \to \frac{π}{2}} \frac{\frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x \cdot 2 - π]}$

Step 3.3.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

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

Step 3.3.3

By the Sum Rule, the derivative of $x \cdot 2 - π$ with respect to $x$ is $\frac{d}{d x} [x \cdot 2] + \frac{d}{d x} [- π]$ .

$2 lim_{x \to \frac{π}{2}} \frac{- \sin (x)}{\frac{d}{d x} [x \cdot 2] + \frac{d}{d x} [- π]}$

Step 3.3.4

Evaluate $\frac{d}{d x} [x \cdot 2]$ .

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Step 3.3.4.1

Move $2$ to the left of $x$ .

$2 lim_{x \to \frac{π}{2}} \frac{- \sin (x)}{\frac{d}{d x} [2 \cdot x] + \frac{d}{d x} [- π]}$

Step 3.3.4.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 lim_{x \to \frac{π}{2}} \frac{- \sin (x)}{2 \frac{d}{d x} [x] + \frac{d}{d x} [- π]}$

Step 3.3.4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 lim_{x \to \frac{π}{2}} \frac{- \sin (x)}{2 \cdot 1 + \frac{d}{d x} [- π]}$

Step 3.3.4.4

Multiply $2$ by $1$ .

$2 lim_{x \to \frac{π}{2}} \frac{- \sin (x)}{2 + \frac{d}{d x} [- π]}$

Step 3.3.5

Since $- π$ is constant with respect to $x$ , the derivative of $- π$ with respect to $x$ is $0$ .

$2 lim_{x \to \frac{π}{2}} \frac{- \sin (x)}{2 + 0}$

Step 3.3.6

Add $2$ and $0$ .

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

Step 4

Evaluate the limit.

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Step 4.1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

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

Step 4.2

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 4.3

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

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

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

Cancel the common factor of $2$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

$1 \cdot - 1 \sin (\frac{π}{2})$

Step 6.2

Multiply $- 1$ by $1$ .

$- \sin (\frac{π}{2})$

Step 6.3

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

$- 1 \cdot 1$

Step 6.4

Multiply $- 1$ by $1$ .

$- 1$