Calculus Examples

$lim_{x \to - 3} 1 - \frac{\sin (\frac{π}{2} - x - 3)}{\sin (2 x + 6)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\frac{\sin (\frac{π}{2} - x - 3)}{\sin (2 x + 6)}$ as a product.

$lim_{x \to - 3} 1 - (\sin (\frac{π}{2} - x - 3) \frac{1}{\sin (2 x + 6)})$

Step 1.2

Write $\sin (\frac{π}{2} - x - 3)$ as a fraction with denominator $1$ .

$lim_{x \to - 3} 1 - (\frac{\sin (\frac{π}{2} - x - 3)}{1} \cdot \frac{1}{\sin (2 x + 6)})$

Step 1.3

Simplify.

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Step 1.3.1

Divide $\sin (\frac{π}{2} - x - 3)$ by $1$ .

$lim_{x \to - 3} 1 - (\sin (\frac{π}{2} - x - 3) \frac{1}{\sin (2 x + 6)})$

Step 1.3.2

Convert from $\frac{1}{\sin (2 x + 6)}$ to $\csc (2 x + 6)$ .

$lim_{x \to - 3} 1 - (\sin (\frac{π}{2} - x - 3) \csc (2 x + 6))$

Step 2

Combine terms.

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Step 2.1

To write $- x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to - 3} 1 - (\sin (\frac{π}{2} - x \cdot \frac{2}{2} - 3) \csc (2 x + 6))$

Step 2.2

Combine $- x$ and $\frac{2}{2}$ .

$lim_{x \to - 3} 1 - (\sin (\frac{π}{2} + \frac{- x \cdot 2}{2} - 3) \csc (2 x + 6))$

Step 2.3

Combine the numerators over the common denominator.

$lim_{x \to - 3} 1 - (\sin (\frac{π - x \cdot 2}{2} - 3) \csc (2 x + 6))$

Step 2.4

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to - 3} 1 - (\sin (\frac{π - x \cdot 2}{2} - 3 \cdot \frac{2}{2}) \csc (2 x + 6))$

Step 2.5

Combine $- 3$ and $\frac{2}{2}$ .

$lim_{x \to - 3} 1 - (\sin (\frac{π - x \cdot 2}{2} + \frac{- 3 \cdot 2}{2}) \csc (2 x + 6))$

Step 2.6

Combine the numerators over the common denominator.

$lim_{x \to - 3} 1 - (\sin (\frac{π - x \cdot 2 - 3 \cdot 2}{2}) \csc (2 x + 6))$

Step 3

Combine factors.

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Step 3.1

Multiply $2$ by $- 1$ .

$lim_{x \to - 3} 1 - (\sin (\frac{π - 2 x - 3 \cdot 2}{2}) \csc (2 x + 6))$

Step 3.2

Multiply $- 3$ by $2$ .

$lim_{x \to - 3} 1 - (\sin (\frac{π - 2 x - 6}{2}) \csc (2 x + 6))$

Step 4

Consider the left sided limit.

$lim_{x \to {(- 3)}^{-}} 1 - (\sin (\frac{π - 2 x - 6}{2}) \csc (2 x + 6))$

Step 5

As the $x$ values approach $- 3$ from the left, the function values increase without bound.

$\infty$

Step 6

Consider the right sided limit.

$lim_{x \to {(- 3)}^{+}} 1 - (\sin (\frac{π - 2 x - 6}{2}) \csc (2 x + 6))$

Step 7

As the $x$ values approach $- 3$ from the right, the function values decrease without bound.

$- \infty$

Step 8

Since the left sided and right sided limits are not equal, the limit does not exist.

$Does not exist$