Calculus Examples

$lim_{x \to - \frac{π}{2}} \frac{\cos (8 x)}{2 x - \sin (\cos (x))}$

Step 1

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

$\frac{lim_{x \to - \frac{π}{2}} \cos (8 x)}{lim_{x \to - \frac{π}{2}} 2 x - \sin (\cos (x))}$

Step 2

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

$\frac{\cos (lim_{x \to - \frac{π}{2}} 8 x)}{lim_{x \to - \frac{π}{2}} 2 x - \sin (\cos (x))}$

Step 3

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{\cos (8 lim_{x \to - \frac{π}{2}} x)}{lim_{x \to - \frac{π}{2}} 2 x - \sin (\cos (x))}$

Step 4

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

$\frac{\cos (8 lim_{x \to - \frac{π}{2}} x)}{lim_{x \to - \frac{π}{2}} 2 x - lim_{x \to - \frac{π}{2}} \sin (\cos (x))}$

Step 5

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

$\frac{\cos (8 lim_{x \to - \frac{π}{2}} x)}{2 lim_{x \to - \frac{π}{2}} x - lim_{x \to - \frac{π}{2}} \sin (\cos (x))}$

Step 6

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

$\frac{\cos (8 lim_{x \to - \frac{π}{2}} x)}{2 lim_{x \to - \frac{π}{2}} x - \sin (lim_{x \to - \frac{π}{2}} \cos (x))}$

Step 7

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

$\frac{\cos (8 lim_{x \to - \frac{π}{2}} x)}{2 lim_{x \to - \frac{π}{2}} x - \sin (\cos (lim_{x \to - \frac{π}{2}} x))}$

Step 8

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

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Step 8.1

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

$\frac{\cos (8 (- \frac{π}{2}))}{2 lim_{x \to - \frac{π}{2}} x - \sin (\cos (lim_{x \to - \frac{π}{2}} x))}$

Step 8.2

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

$\frac{\cos (8 (- \frac{π}{2}))}{2 (- \frac{π}{2}) - \sin (\cos (lim_{x \to - \frac{π}{2}} x))}$

Step 8.3

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

$\frac{\cos (8 (- \frac{π}{2}))}{2 (- \frac{π}{2}) - \sin (\cos (- \frac{π}{2}))}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Cancel the common factor of $2$ .

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Step 9.1.1.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$\frac{\cos (8 \frac{- π}{2})}{2 (- \frac{π}{2}) - \sin (\cos (- \frac{π}{2}))}$

Step 9.1.1.2

Factor $2$ out of $8$ .

$\frac{\cos (2 (4) \frac{- π}{2})}{2 (- \frac{π}{2}) - \sin (\cos (- \frac{π}{2}))}$

Step 9.1.1.3

Cancel the common factor.

$\frac{\cos (2 \cdot 4 \frac{- π}{2})}{2 (- \frac{π}{2}) - \sin (\cos (- \frac{π}{2}))}$

Step 9.1.1.4

Rewrite the expression.

$\frac{\cos (4 (- π))}{2 (- \frac{π}{2}) - \sin (\cos (- \frac{π}{2}))}$

Step 9.1.2

Multiply $- 1$ by $4$ .

$\frac{\cos (- 4 π)}{2 (- \frac{π}{2}) - \sin (\cos (- \frac{π}{2}))}$

Step 9.1.3

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

$\frac{\cos (0)}{2 (- \frac{π}{2}) - \sin (\cos (- \frac{π}{2}))}$

Step 9.1.4

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

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

Step 9.2

Simplify the denominator.

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Step 9.2.1

Cancel the common factor of $2$ .

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Step 9.2.1.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

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

Step 9.2.1.2

Cancel the common factor.

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

Step 9.2.1.3

Rewrite the expression.

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

Step 9.2.2

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

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

Step 9.2.3

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

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

Step 9.2.4

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

$\frac{1}{- π - \sin (0)}$

Step 9.2.5

The exact value of $\sin (0)$ is $0$ .

$\frac{1}{- π - 0}$

Step 9.2.6

Multiply $- 1$ by $0$ .

$\frac{1}{- π + 0}$

Step 9.2.7

Add $- π$ and $0$ .

$\frac{1}{- π}$

Step 9.3

Cancel the common factor of $1$ and $- 1$ .

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Step 9.3.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- π}$

Step 9.3.2

Move the negative in front of the fraction.

$- \frac{1}{π}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{π}$

Decimal Form:

$- 0.31830988 \dots$