Calculus Examples

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

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.

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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Step 1.1.2.4.1

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

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

Step 1.1.2.4.2

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

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

Step 1.1.2.5

Simplify the answer.

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Step 1.1.2.5.1

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

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

Step 1.1.2.5.2

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

$\frac{0 \cdot 1}{lim_{x \to 0} x}$

Step 1.1.2.5.3

Multiply $0$ by $1$ .

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

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin (x)$ and $g (x) = \cos (x)$ .

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

Step 1.3.3

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

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

Step 1.3.4

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.3.5

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.3.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.3.7

Add $1$ and $1$ .

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

Step 1.3.8

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

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

Step 1.3.9

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.3.10

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.3.11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.3.12

Add $1$ and $1$ .

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

Step 1.3.13

Simplify.

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Step 1.3.13.1

Reorder $- \sin^{2} (x)$ and $\cos^{2} (x)$ .

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

Step 1.3.13.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (x)$ and $b = \sin (x)$ .

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

Step 1.3.13.3

Expand $(\cos (x) + \sin (x)) (\cos (x) - \sin (x))$ using the FOIL Method.

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Step 1.3.13.3.1

Apply the distributive property.

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

Step 1.3.13.3.2

Apply the distributive property.

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

Step 1.3.13.3.3

Apply the distributive property.

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

Step 1.3.13.4

Combine the opposite terms in $\cos (x) \cos (x) + \cos (x) (- \sin (x)) + \sin (x) \cos (x) + \sin (x) (- \sin (x))$ .

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Step 1.3.13.4.1

Reorder the factors in the terms $\cos (x) (- \sin (x))$ and $\sin (x) \cos (x)$ .

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

Step 1.3.13.4.2

Add $- \cos (x) \sin (x)$ and $\cos (x) \sin (x)$ .

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

Step 1.3.13.4.3

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

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

Step 1.3.13.5

Simplify each term.

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Step 1.3.13.5.1

Multiply $\cos (x) \cos (x)$ .

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Step 1.3.13.5.1.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.3.13.5.1.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.3.13.5.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.3.13.5.1.4

Add $1$ and $1$ .

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

Step 1.3.13.5.2

Rewrite using the commutative property of multiplication.

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

Step 1.3.13.5.3

Multiply $- \sin (x) \sin (x)$ .

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Step 1.3.13.5.3.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.3.13.5.3.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.3.13.5.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.3.13.5.3.4

Add $1$ and $1$ .

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

Step 1.3.13.6

Apply the cosine double-angle identity.

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

Step 1.3.14

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

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

Step 1.4

Divide $\cos (2 x)$ by $1$ .

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

$\cos (2 \cdot 0)$

Step 4

Simplify the answer.

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Step 4.1

Multiply $2$ by $0$ .

$\cos (0)$

Step 4.2

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

$1$