Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $π$ .

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

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $π$ .

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

Step 3

Evaluate the limit of $1$ which is constant as $x$ approaches $π$ .

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

Step 4

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

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

Step 5

Evaluate the limits by plugging in $π$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $π$ for $x$ .

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

Step 5.2

Evaluate the limit of $x$ by plugging in $π$ for $x$ .

$\frac{1 - \cos (π)}{π}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{1 - - \cos (0)}{π}$

Step 6.2

The exact value of $\cos (0)$ is $1$ .

$\frac{1 - (- 1 \cdot 1)}{π}$

Step 6.3

Multiply $- (- 1 \cdot 1)$ .

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Step 6.3.1

Multiply $- 1$ by $1$ .

$\frac{1 - - 1}{π}$

Step 6.3.2

Multiply $- 1$ by $- 1$ .

$\frac{1 + 1}{π}$

Step 6.4

Add $1$ and $1$ .

$\frac{2}{π}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{π}$

Decimal Form:

$0.63661977 \dots$