Calculus Examples

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

Step 1

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

$3 \frac{0}{lim_{x \to \frac{π}{2}} 2 x - π}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Evaluate the limit.

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Step 2.1.3.1.1

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

$3 \frac{0}{lim_{x \to \frac{π}{2}} 2 x - lim_{x \to \frac{π}{2}} π}$

Step 2.1.3.1.2

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

$3 \frac{0}{2 lim_{x \to \frac{π}{2}} x - lim_{x \to \frac{π}{2}} π}$

Step 2.1.3.1.3

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

$3 \frac{0}{2 lim_{x \to \frac{π}{2}} x - π}$

Step 2.1.3.2

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

$3 \frac{0}{2 \frac{π}{2} - π}$

Step 2.1.3.3

Simplify the answer.

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Step 2.1.3.3.1

Cancel the common factor of $2$ .

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Step 2.1.3.3.1.1

Cancel the common factor.

$3 \frac{0}{2 \frac{π}{2} - π}$

Step 2.1.3.3.1.2

Rewrite the expression.

$3 \frac{0}{π - π}$

Step 2.1.3.3.2

Subtract $π$ from $π$ .

$3 (\frac{0}{0})$

Step 2.1.3.3.3

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

Undefined

$3 (\frac{0}{0})$

Step 2.1.3.4

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

Undefined

$3 (\frac{0}{0})$

Step 2.1.4

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

Undefined

$3 (\frac{0}{0})$

Step 2.2

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$3 lim_{x \to \frac{π}{2}} \frac{\frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [2 x - π]}$

Step 2.3.2

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

$3 lim_{x \to \frac{π}{2}} \frac{- \sin (x)}{\frac{d}{d x} [2 x - π]}$

Step 2.3.3

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

$3 lim_{x \to \frac{π}{2}} \frac{- \sin (x)}{\frac{d}{d x} [2 x] + \frac{d}{d x} [- π]}$

Step 2.3.4

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

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Step 2.3.4.1

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

$3 lim_{x \to \frac{π}{2}} \frac{- \sin (x)}{2 \frac{d}{d x} [x] + \frac{d}{d x} [- π]}$

Step 2.3.4.2

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

$3 lim_{x \to \frac{π}{2}} \frac{- \sin (x)}{2 \cdot 1 + \frac{d}{d x} [- π]}$

Step 2.3.4.3

Multiply $2$ by $1$ .

$3 lim_{x \to \frac{π}{2}} \frac{- \sin (x)}{2 + \frac{d}{d x} [- π]}$

Step 2.3.5

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

$3 lim_{x \to \frac{π}{2}} \frac{- \sin (x)}{2 + 0}$

Step 2.3.6

Add $2$ and $0$ .

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

Step 3

Evaluate the limit.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

Combine $3$ and $\frac{1}{2}$ .

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

Step 5.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 5.2.1

Combine $\frac{3}{2}$ and $- 1$ .

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

Step 5.2.2

Multiply $3$ by $- 1$ .

$\frac{- 3}{2} \sin (\frac{π}{2})$

Step 5.3

Move the negative in front of the fraction.

$- \frac{3}{2} \sin (\frac{π}{2})$

Step 5.4

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

$- \frac{3}{2} \cdot 1$

Step 5.5

Multiply $- 1$ by $1$ .

$- \frac{3}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{2}$

Decimal Form:

$- 1.5$