Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

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 $x$ .

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

Step 5.2

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

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$\frac{0 - \cos (3 \frac{π}{2})}{(2 x - π) (\sqrt{\cos (x)} + \sqrt{\cos (3 x)})}$

Step 6.1.2

Combine $3$ and $\frac{π}{2}$ .

$\frac{0 - \cos (\frac{3 π}{2})}{(2 x - π) (\sqrt{\cos (x)} + \sqrt{\cos (3 x)})}$

Step 6.1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{0 - \cos (\frac{π}{2})}{(2 x - π) (\sqrt{\cos (x)} + \sqrt{\cos (3 x)})}$

Step 6.1.4

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

$\frac{0 - 0}{(2 x - π) (\sqrt{\cos (x)} + \sqrt{\cos (3 x)})}$

Step 6.1.5

Multiply $- 1$ by $0$ .

$\frac{0 + 0}{(2 x - π) (\sqrt{\cos (x)} + \sqrt{\cos (3 x)})}$

Step 6.1.6

Add $0$ and $0$ .

$\frac{0}{(2 x - π) (\sqrt{\cos (x)} + \sqrt{\cos (3 x)})}$

Step 6.2

Divide $0$ by $(2 x - π) (\sqrt{\cos (x)} + \sqrt{\cos (3 x)})$ .

$0$