Calculus Examples

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

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{1}$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Evaluate the limits by plugging in $\frac{π}{1}$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $\frac{π}{1}$ for $x$ .

$\frac{1 - 2 \cos (\frac{π}{1})}{π - 3 x}$

Step 5.2

Divide $π$ by $1$ .

$\frac{1 - 2 \cos (π)}{π - 3 x}$

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.

$\frac{1 - 2 (- \cos (0))}{π - 3 x}$

Step 6.2

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

$\frac{1 - 2 (- 1 \cdot 1)}{π - 3 x}$

Step 6.3

Multiply $- 2 (- 1 \cdot 1)$ .

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Step 6.3.1

Multiply $- 1$ by $1$ .

$\frac{1 - 2 \cdot - 1}{π - 3 x}$

Step 6.3.2

Multiply $- 2$ by $- 1$ .

$\frac{1 + 2}{π - 3 x}$

Step 6.4

Add $1$ and $2$ .

$\frac{3}{π - 3 x}$