Calculus Examples

$lim_{x \to 0} \frac{{(1 - \cos (x))}^{2}}{x}$

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.

$\frac{lim_{x \to 0} {(1 - \cos (x))}^{2}}{lim_{x \to 0} x}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Move the exponent $2$ from ${(1 - \cos (x))}^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 0} 1 - \cos (x))}^{2}}{lim_{x \to 0} x}$

Step 1.1.2.1.2

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

$\frac{{(lim_{x \to 0} 1 - lim_{x \to 0} \cos (x))}^{2}}{lim_{x \to 0} x}$

Step 1.1.2.1.3

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

$\frac{{(1 - lim_{x \to 0} \cos (x))}^{2}}{lim_{x \to 0} x}$

Step 1.1.2.1.4

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

$\frac{{(1 - \cos (lim_{x \to 0} x))}^{2}}{lim_{x \to 0} x}$

Step 1.1.2.2

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

$\frac{{(1 - \cos (0))}^{2}}{lim_{x \to 0} x}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

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

$\frac{{(1 - 1 \cdot 1)}^{2}}{lim_{x \to 0} x}$

Step 1.1.2.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0^{2}}{lim_{x \to 0} x}$

Step 1.1.2.3.3

Raising $0$ to any positive power yields $0$ .

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

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.

$lim_{x \to 0} \frac{{(1 - \cos (x))}^{2}}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [{(1 - \cos (x))}^{2}]}{\frac{d}{d x} [x]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [{(1 - \cos (x))}^{2}]}{\frac{d}{d x} [x]}$

Step 1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 1 - \cos (x)$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u$ as $1 - \cos (x)$ .

$lim_{x \to 0} \frac{\frac{d}{d u} [u^{2}] \frac{d}{d x} [1 - \cos (x)]}{\frac{d}{d x} [x]}$

Step 1.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$lim_{x \to 0} \frac{2 u \frac{d}{d x} [1 - \cos (x)]}{\frac{d}{d x} [x]}$

Step 1.3.2.3

Replace all occurrences of $u$ with $1 - \cos (x)$ .

$lim_{x \to 0} \frac{2 (1 - \cos (x)) \frac{d}{d x} [1 - \cos (x)]}{\frac{d}{d x} [x]}$

Step 1.3.3

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

$lim_{x \to 0} \frac{2 (1 - \cos (x)) (\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (x)])}{\frac{d}{d x} [x]}$

Step 1.3.4

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

$lim_{x \to 0} \frac{2 (1 - \cos (x)) (0 + \frac{d}{d x} [- \cos (x)])}{\frac{d}{d x} [x]}$

Step 1.3.5

Add $0$ and $\frac{d}{d x} [- \cos (x)]$ .

$lim_{x \to 0} \frac{2 (1 - \cos (x)) \frac{d}{d x} [- \cos (x)]}{\frac{d}{d x} [x]}$

Step 1.3.6

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

$lim_{x \to 0} \frac{2 (1 - \cos (x)) (- \frac{d}{d x} [\cos (x)])}{\frac{d}{d x} [x]}$

Step 1.3.7

Multiply $- 1$ by $2$ .

$lim_{x \to 0} \frac{- 2 (1 - \cos (x)) \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x]}$

Step 1.3.8

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

$lim_{x \to 0} \frac{- 2 (1 - \cos (x)) (- \sin (x))}{\frac{d}{d x} [x]}$

Step 1.3.9

Multiply $- 1$ by $- 2$ .

$lim_{x \to 0} \frac{2 (1 - \cos (x)) \sin (x)}{\frac{d}{d x} [x]}$

Step 1.3.10

Simplify.

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Step 1.3.10.1

Apply the distributive property.

$lim_{x \to 0} \frac{(2 \cdot 1 + 2 (- \cos (x))) \sin (x)}{\frac{d}{d x} [x]}$

Step 1.3.10.2

Apply the distributive property.

$lim_{x \to 0} \frac{2 \cdot 1 \sin (x) + 2 (- \cos (x)) \sin (x)}{\frac{d}{d x} [x]}$

Step 1.3.10.3

Combine terms.

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Step 1.3.10.3.1

Multiply $2$ by $1$ .

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

Step 1.3.10.3.2

Multiply $- 1$ by $2$ .

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

Step 1.3.10.4

Reorder terms.

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

Step 1.3.11

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

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

Step 1.4

Divide $- 2 \cos (x) \sin (x) + 2 \sin (x)$ by $1$ .

$lim_{x \to 0} - 2 \cos (x) \sin (x) + 2 \sin (x)$

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 $0$ .

$- lim_{x \to 0} 2 \cos (x) \sin (x) + lim_{x \to 0} 2 \sin (x)$

Step 2.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$- 2 lim_{x \to 0} \cos (x) \sin (x) + lim_{x \to 0} 2 \sin (x)$

Step 2.3

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

$- 2 lim_{x \to 0} \cos (x) \cdot lim_{x \to 0} \sin (x) + lim_{x \to 0} 2 \sin (x)$

Step 2.4

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

$- 2 \cos (lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (x) + lim_{x \to 0} 2 \sin (x)$

Step 2.5

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

$- 2 \cos (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) + lim_{x \to 0} 2 \sin (x)$

Step 2.6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$- 2 \cos (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) + 2 lim_{x \to 0} \sin (x)$

Step 2.7

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

$- 2 \cos (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) + 2 \sin (lim_{x \to 0} x)$

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 $x$ .

$- 2 \cos (0) \cdot \sin (lim_{x \to 0} x) + 2 \sin (lim_{x \to 0} x)$

Step 3.2

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

$- 2 \cos (0) \cdot \sin (0) + 2 \sin (lim_{x \to 0} x)$

Step 3.3

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

$- 2 \cos (0) \cdot \sin (0) + 2 \sin (0)$

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

$- 2 \cdot 1 \cdot \sin (0) + 2 \sin (0)$

Step 4.1.2

Multiply $- 2$ by $1$ .

$- 2 \cdot \sin (0) + 2 \sin (0)$

Step 4.1.3

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

$- 2 \cdot 0 + 2 \sin (0)$

Step 4.1.4

Multiply $- 2$ by $0$ .

$0 + 2 \sin (0)$

Step 4.1.5

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

$0 + 2 \cdot 0$

Step 4.1.6

Multiply $2$ by $0$ .

$0 + 0$

Step 4.2

Add $0$ and $0$ .

$0$