Calculus Examples

$lim_{x \to - \frac{π}{2}} \cos (5 x - \cos (5 x))$

Step 1

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

$\cos (lim_{x \to - \frac{π}{2}} 5 x - \cos (5 x))$

Step 2

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

$\cos (lim_{x \to - \frac{π}{2}} 5 x - lim_{x \to - \frac{π}{2}} \cos (5 x))$

Step 3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\cos (5 lim_{x \to - \frac{π}{2}} x - lim_{x \to - \frac{π}{2}} \cos (5 x))$

Step 4

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

$\cos (5 lim_{x \to - \frac{π}{2}} x - \cos (lim_{x \to - \frac{π}{2}} 5 x))$

Step 5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\cos (5 lim_{x \to - \frac{π}{2}} x - \cos (5 lim_{x \to - \frac{π}{2}} x))$

Step 6

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

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Step 6.1

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

$\cos (5 (- \frac{π}{2}) - \cos (5 lim_{x \to - \frac{π}{2}} x))$

Step 6.2

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

$\cos (5 (- \frac{π}{2}) - \cos (5 (- \frac{π}{2})))$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $5 (- \frac{π}{2})$ .

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Step 7.1.1.1

Multiply $- 1$ by $5$ .

$\cos (- 5 \frac{π}{2} - \cos (5 (- \frac{π}{2})))$

Step 7.1.1.2

Combine $- 5$ and $\frac{π}{2}$ .

$\cos (\frac{- 5 π}{2} - \cos (5 (- \frac{π}{2})))$

Step 7.1.2

Move the negative in front of the fraction.

$\cos (- \frac{5 π}{2} - \cos (5 (- \frac{π}{2})))$

Step 7.1.3

Multiply $5 (- \frac{π}{2})$ .

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Step 7.1.3.1

Multiply $- 1$ by $5$ .

$\cos (- \frac{5 π}{2} - \cos (- 5 \frac{π}{2}))$

Step 7.1.3.2

Combine $- 5$ and $\frac{π}{2}$ .

$\cos (- \frac{5 π}{2} - \cos (\frac{- 5 π}{2}))$

Step 7.1.4

Move the negative in front of the fraction.

$\cos (- \frac{5 π}{2} - \cos (- \frac{5 π}{2}))$

Step 7.1.5

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\cos (- \frac{5 π}{2} - \cos (\frac{3 π}{2}))$

Step 7.1.6

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

$\cos (- \frac{5 π}{2} - \cos (\frac{π}{2}))$

Step 7.1.7

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

$\cos (- \frac{5 π}{2} - 0)$

Step 7.1.8

Multiply $- 1$ by $0$ .

$\cos (- \frac{5 π}{2} + 0)$

Step 7.2

Add $- \frac{5 π}{2}$ and $0$ .

$\cos (- \frac{5 π}{2})$

Step 7.3

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\cos (\frac{3 π}{2})$

Step 7.4

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

$\cos (\frac{π}{2})$

Step 7.5

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

$0$