Calculus Examples

$lim_{x \to \frac{π}{2}} \frac{1 - 8 \cos (x)}{2 x}$

Step 1

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

$\frac{1}{2} lim_{x \to \frac{π}{2}} \frac{1 - 8 \cos (x)}{x}$

Step 2

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

$\frac{1}{2} \cdot \frac{lim_{x \to \frac{π}{2}} 1 - 8 \cos (x)}{lim_{x \to \frac{π}{2}} x}$

Step 3

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

$\frac{1}{2} \cdot \frac{lim_{x \to \frac{π}{2}} 1 - lim_{x \to \frac{π}{2}} 8 \cos (x)}{lim_{x \to \frac{π}{2}} x}$

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{π}{2}$ .

$\frac{1}{2} \cdot \frac{1 - lim_{x \to \frac{π}{2}} 8 \cos (x)}{lim_{x \to \frac{π}{2}} x}$

Step 5

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot \frac{1 - 8 lim_{x \to \frac{π}{2}} \cos (x)}{lim_{x \to \frac{π}{2}} x}$

Step 6

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

$\frac{1}{2} \cdot \frac{1 - 8 \cos (lim_{x \to \frac{π}{2}} x)}{lim_{x \to \frac{π}{2}} x}$

Step 7

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

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Step 7.1

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

$\frac{1}{2} \cdot \frac{1 - 8 \cos (\frac{π}{2})}{lim_{x \to \frac{π}{2}} x}$

Step 7.2

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

$\frac{1}{2} \cdot \frac{1 - 8 \cos (\frac{π}{2})}{\frac{π}{2}}$

Step 8

Simplify the answer.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} ((1 - 8 \cos (\frac{π}{2})) \frac{2}{π})$

Step 8.2

Simplify each term.

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Step 8.2.1

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

$\frac{1}{2} ((1 - 8 \cdot 0) \frac{2}{π})$

Step 8.2.2

Multiply $- 8$ by $0$ .

$\frac{1}{2} ((1 + 0) \frac{2}{π})$

Step 8.3

Add $1$ and $0$ .

$\frac{1}{2} (1 \frac{2}{π})$

Step 8.4

Multiply $\frac{2}{π}$ by $1$ .

$\frac{1}{2} \cdot \frac{2}{π}$

Step 8.5

Cancel the common factor of $2$ .

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Step 8.5.1

Cancel the common factor.

$\frac{1}{2} \cdot \frac{2}{π}$

Step 8.5.2

Rewrite the expression.

$\frac{1}{π}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{π}$

Decimal Form:

$0.31830988 \dots$