Calculus Examples

$\frac{lim_{x \to 0} 1 - \cos (x) + \sin (x)}{1 - \cos (x) - \sin (x)}$

Step 1

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

$\frac{lim_{x \to 0} 1 - lim_{x \to 0} \cos (x) + lim_{x \to 0} \sin (x)}{1 - \cos (x) - \sin (x)}$

Step 2

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

$\frac{1 - lim_{x \to 0} \cos (x) + lim_{x \to 0} \sin (x)}{1 - \cos (x) - \sin (x)}$

Step 3

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

$\frac{1 - \cos (lim_{x \to 0} x) + lim_{x \to 0} \sin (x)}{1 - \cos (x) - \sin (x)}$

Step 4

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

$\frac{1 - \cos (lim_{x \to 0} x) + \sin (lim_{x \to 0} x)}{1 - \cos (x) - \sin (x)}$

Step 5

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 5.1

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

$\frac{1 - \cos (0) + \sin (lim_{x \to 0} x)}{1 - \cos (x) - \sin (x)}$

Step 5.2

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

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

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 $\cos (0)$ is $1$ .

$\frac{1 - 1 \cdot 1 + \sin (0)}{1 - \cos (x) - \sin (x)}$

Step 6.1.2

Multiply $- 1$ by $1$ .

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

Step 6.1.3

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

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

Step 6.1.4

Add $1 - 1$ and $0$ .

$\frac{1 - 1}{1 - \cos (x) - \sin (x)}$

Step 6.1.5

Subtract $1$ from $1$ .

$\frac{0}{1 - \cos (x) - \sin (x)}$

Step 6.2

Divide $0$ by $1 - \cos (x) - \sin (x)$ .

$0$