Calculus Examples

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

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.

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

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 $0$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

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

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

Step 1.1.2.6

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

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

Step 1.1.2.7

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

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

Step 1.1.2.8

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

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Step 1.1.2.8.1

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

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

Step 1.1.2.8.2

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

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

Step 1.1.2.9

Combine the opposite terms in $\cos (a + 0) - \cos (a + 0)$ .

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Step 1.1.2.9.1

Add $a$ and $0$ .

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

Step 1.1.2.9.2

Add $a$ and $0$ .

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

Step 1.1.2.9.3

Subtract $\cos (a)$ from $\cos (a)$ .

$\frac{0}{lim_{x \to 0} x}$

Step 1.1.3

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

$\frac{0}{0}$

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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Step 1.3.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = a + x$ .

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Step 1.3.3.1.1

To apply the Chain Rule, set $u_{1}$ as $a + x$ .

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

Step 1.3.3.1.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

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

Step 1.3.3.1.3

Replace all occurrences of $u_{1}$ with $a + x$ .

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

Step 1.3.3.2

By the Sum Rule, the derivative of $a + x$ with respect to $x$ is $\frac{d}{d x} [a] + \frac{d}{d x} [x]$ .

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

Step 1.3.3.3

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

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

Step 1.3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

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

Step 1.3.3.5

Add $0$ and $1$ .

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

Step 1.3.3.6

Multiply $- 1$ by $1$ .

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

Step 1.3.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = a - x$ .

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Step 1.3.4.2.1

To apply the Chain Rule, set $u_{2}$ as $a - x$ .

$lim_{x \to 0} \frac{- \sin (a + x) - (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [a - x])}{\frac{d}{d x} [x]}$

Step 1.3.4.2.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$lim_{x \to 0} \frac{- \sin (a + x) - (- \sin (u_{2}) \frac{d}{d x} [a - x])}{\frac{d}{d x} [x]}$

Step 1.3.4.2.3

Replace all occurrences of $u_{2}$ with $a - x$ .

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

Step 1.3.4.3

By the Sum Rule, the derivative of $a - x$ with respect to $x$ is $\frac{d}{d x} [a] + \frac{d}{d x} [- x]$ .

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

Step 1.3.4.4

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

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

Step 1.3.4.5

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

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

Step 1.3.4.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

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

Step 1.3.4.7

Multiply $- 1$ by $1$ .

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

Step 1.3.4.8

Subtract $1$ from $0$ .

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

Step 1.3.4.9

Multiply $- 1$ by $- 1$ .

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

Step 1.3.4.10

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

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

Step 1.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

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

Step 1.4

Divide $- \sin (a + x) - \sin (a - x)$ by $1$ .

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

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 $x$ .

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

Step 3.2

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

$- \sin (a + 0) - \sin (a + 0)$

Step 4

Simplify the answer.

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Step 4.1

Combine the opposite terms in $- \sin (a + 0) - \sin (a + 0)$ .

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Step 4.1.1

Add $a$ and $0$ .

$- \sin (a) - \sin (a + 0)$

Step 4.1.2

Add $a$ and $0$ .

$- \sin (a) - \sin (a)$

Step 4.2

Subtract $\sin (a)$ from $- \sin (a)$ .

$- 2 \sin (a)$