Calculus Examples

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

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

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

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Step 1.1.2.5.1

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

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

Step 1.1.2.5.2

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

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

Step 1.1.2.6

Simplify the answer.

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Step 1.1.2.6.1

Simplify each term.

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Step 1.1.2.6.1.1

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

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

Step 1.1.2.6.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.2.6.1.3

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

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

Step 1.1.2.6.2

Subtract $1$ from $1$ .

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

Step 1.1.2.6.3

Add $0$ and $0$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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

Step 1.1.3.5

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

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Step 1.1.3.5.1

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

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

Step 1.1.3.5.2

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

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

Step 1.1.3.6

Simplify the answer.

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Step 1.1.3.6.1

Simplify each term.

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Step 1.1.3.6.1.1

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

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

Step 1.1.3.6.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.3.6.1.3

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

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

Step 1.1.3.6.1.4

Multiply $- 1$ by $0$ .

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

Step 1.1.3.6.2

Subtract $1$ from $1$ .

$\frac{0}{0 + 0}$

Step 1.1.3.6.3

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.1.3.6.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.7

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

By the Sum Rule, the derivative of $1 - \cos (x) + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (x)] + \frac{d}{d x} [\sin (x)]$ .

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

Step 1.3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.3.4

Evaluate $\frac{d}{d x} [- \cos (x)]$ .

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Step 1.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x)]$ .

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

Step 1.3.4.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

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

Step 1.3.4.3

Multiply $- 1$ by $- 1$ .

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

Step 1.3.4.4

Multiply $\sin (x)$ by $1$ .

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

Step 1.3.5

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

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

Step 1.3.6

Add $0$ and $\sin (x)$ .

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

Step 1.3.7

By the Sum Rule, the derivative of $1 - \cos (x) - \sin (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (x)] + \frac{d}{d x} [- \sin (x)]$ .

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

Step 1.3.8

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{\sin (x) + \cos (x)}{0 + \frac{d}{d x} [- \cos (x)] + \frac{d}{d x} [- \sin (x)]}$

Step 1.3.9

Evaluate $\frac{d}{d x} [- \cos (x)]$ .

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Step 1.3.9.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x)]$ .

$lim_{x \to 0} \frac{\sin (x) + \cos (x)}{0 - \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- \sin (x)]}$

Step 1.3.9.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$lim_{x \to 0} \frac{\sin (x) + \cos (x)}{0 - - \sin (x) + \frac{d}{d x} [- \sin (x)]}$

Step 1.3.9.3

Multiply $- 1$ by $- 1$ .

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

Step 1.3.9.4

Multiply $\sin (x)$ by $1$ .

$lim_{x \to 0} \frac{\sin (x) + \cos (x)}{0 + \sin (x) + \frac{d}{d x} [- \sin (x)]}$

Step 1.3.10

Evaluate $\frac{d}{d x} [- \sin (x)]$ .

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Step 1.3.10.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (x)$ with respect to $x$ is $- \frac{d}{d x} [\sin (x)]$ .

$lim_{x \to 0} \frac{\sin (x) + \cos (x)}{0 + \sin (x) - \frac{d}{d x} [\sin (x)]}$

Step 1.3.10.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

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

Step 1.3.11

Add $0$ and $\sin (x)$ .

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

Add $0$ and $1$ .

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

Step 4.2

Simplify the denominator.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

Multiply $- 1$ by $1$ .

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

Step 4.2.4

Subtract $1$ from $0$ .

$\frac{1}{- 1}$

Step 4.3

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

$- 1 \cdot 1$

Step 4.4

Multiply $- 1$ by $1$ .

$- 1$