Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

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

Step 9.1.2

Multiply $- 1$ by $0$ .

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

Step 9.1.3

Add $1$ and $0$ .

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

Step 9.2

Simplify the denominator.

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Step 9.2.1

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

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

Step 9.2.2

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

$\frac{1}{0 - 1 \cdot 1}$

Step 9.2.3

Multiply $- 1$ by $1$ .

$\frac{1}{0 - 1}$

Step 9.2.4

Subtract $1$ from $0$ .

$\frac{1}{- 1}$

Step 9.3

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$- 1 \cdot 1$

Step 9.4

Multiply $- 1$ by $1$ .

$- 1$