Calculus Examples

$\frac{lim_{x \to 0} 1 - \sin ((\frac{π}{2}) - x)}{x}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 1 - lim_{x \to 0} \sin ((\frac{π}{2}) - x)}{x}$

Step 2

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{1 - lim_{x \to 0} \sin ((\frac{π}{2}) - x)}{x}$

Step 3

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

$\frac{1 - \sin (lim_{x \to 0} (\frac{π}{2}) - x)}{x}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1 - \sin (lim_{x \to 0} \frac{π}{2} - lim_{x \to 0} x)}{x}$

Step 5

Evaluate the limit of $\frac{π}{2}$ which is constant as $x$ approaches $0$ .

$\frac{1 - \sin (\frac{π}{2} - lim_{x \to 0} x)}{x}$

Step 6

Simplify terms.

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Step 6.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1 - \sin (\frac{π}{2} + 0)}{x}$

Step 6.2

Simplify the answer.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Add $\frac{π}{2}$ and $0$ .

$\frac{1 - \sin (\frac{π}{2})}{x}$

Step 6.2.1.2

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{1 - 1 \cdot 1}{x}$

Step 6.2.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{x}$

Step 6.2.1.4

Subtract $1$ from $1$ .

$\frac{0}{x}$

Step 6.2.2

Divide $0$ by $x$ .

$0$