Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Simplify the answer.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.2

Simplify each term.

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Step 6.2.1

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

$(1 - 0) \frac{2}{π}$

Step 6.2.2

Multiply $- 1$ by $0$ .

$(1 + 0) \frac{2}{π}$

Step 6.3

Add $1$ and $0$ .

$1 \frac{2}{π}$

Step 6.4

Multiply $\frac{2}{π}$ by $1$ .

$\frac{2}{π}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{π}$

Decimal Form:

$0.63661977 \dots$