Calculus Examples

$lim_{x \to 0} \frac{\sin (3 x) \cos (x)}{x}$

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.

$\frac{lim_{x \to 0} \sin (3 x) \cos (x)}{lim_{x \to 0} x}$

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 $0$ .

$\frac{lim_{x \to 0} \sin (3 x) \cdot lim_{x \to 0} \cos (x)}{lim_{x \to 0} x}$

Step 1.1.2.2

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

$\frac{\sin (lim_{x \to 0} 3 x) \cdot lim_{x \to 0} \cos (x)}{lim_{x \to 0} x}$

Step 1.1.2.3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (3 lim_{x \to 0} x) \cdot lim_{x \to 0} \cos (x)}{lim_{x \to 0} x}$

Step 1.1.2.4

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

$\frac{\sin (3 lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x)}{lim_{x \to 0} x}$

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 $x$ .

$\frac{\sin (3 \cdot 0) \cdot \cos (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 1.1.2.5.2

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

$\frac{\sin (3 \cdot 0) \cdot \cos (0)}{lim_{x \to 0} x}$

Step 1.1.2.6

Simplify the answer.

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Step 1.1.2.6.1

Multiply $3$ by $0$ .

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

Step 1.1.2.6.2

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

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

Step 1.1.2.6.3

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

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

Step 1.1.2.6.4

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.

$lim_{x \to 0} \frac{\sin (3 x) \cos (x)}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (3 x) \cos (x)]}{\frac{d}{d x} [x]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\sin (3 x) \cos (x)]}{\frac{d}{d x} [x]}$

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 $f (x) = \sin (3 x)$ and $g (x) = \cos (x)$ .

$lim_{x \to 0} \frac{\sin (3 x) \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [\sin (3 x)]}{\frac{d}{d x} [x]}$

Step 1.3.3

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

$lim_{x \to 0} \frac{\sin (3 x) (- \sin (x)) + \cos (x) \frac{d}{d x} [\sin (3 x)]}{\frac{d}{d x} [x]}$

Step 1.3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 3 x$ .

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Step 1.3.4.1

To apply the Chain Rule, set $u$ as $3 x$ .

$lim_{x \to 0} \frac{\sin (3 x) (- \sin (x)) + \cos (x) (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [3 x])}{\frac{d}{d x} [x]}$

Step 1.3.4.2

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

$lim_{x \to 0} \frac{\sin (3 x) (- \sin (x)) + \cos (x) (\cos (u) \frac{d}{d x} [3 x])}{\frac{d}{d x} [x]}$

Step 1.3.4.3

Replace all occurrences of $u$ with $3 x$ .

$lim_{x \to 0} \frac{\sin (3 x) (- \sin (x)) + \cos (x) (\cos (3 x) \frac{d}{d x} [3 x])}{\frac{d}{d x} [x]}$

Step 1.3.5

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\sin (3 x) (- \sin (x)) + \cos (x) \cos (3 x) (3 \frac{d}{d x} [x])}{\frac{d}{d x} [x]}$

Step 1.3.6

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

$lim_{x \to 0} \frac{\sin (3 x) (- \sin (x)) + \cos (x) \cos (3 x) (3 \cdot 1)}{\frac{d}{d x} [x]}$

Step 1.3.7

Multiply $3$ by $1$ .

$lim_{x \to 0} \frac{\sin (3 x) (- \sin (x)) + \cos (x) \cos (3 x) \cdot 3}{\frac{d}{d x} [x]}$

Step 1.3.8

Move $3$ to the left of $\cos (x) \cos (3 x)$ .

$lim_{x \to 0} \frac{\sin (3 x) (- \sin (x)) + 3 \cdot (\cos (x) \cos (3 x))}{\frac{d}{d x} [x]}$

Step 1.3.9

Reorder terms.

$lim_{x \to 0} \frac{- \sin (x) \sin (3 x) + 3 \cos (x) \cos (3 x)}{\frac{d}{d x} [x]}$

Step 1.3.10

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

$lim_{x \to 0} \frac{- \sin (x) \sin (3 x) + 3 \cos (x) \cos (3 x)}{1}$

Step 1.4

Divide $- \sin (x) \sin (3 x) + 3 \cos (x) \cos (3 x)$ by $1$ .

$lim_{x \to 0} - \sin (x) \sin (3 x) + 3 \cos (x) \cos (3 x)$

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 $0$ .

$- lim_{x \to 0} \sin (x) \sin (3 x) + lim_{x \to 0} 3 \cos (x) \cos (3 x)$

Step 2.2

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

$- lim_{x \to 0} \sin (x) \cdot lim_{x \to 0} \sin (3 x) + lim_{x \to 0} 3 \cos (x) \cos (3 x)$

Step 2.3

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

$- \sin (lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (3 x) + lim_{x \to 0} 3 \cos (x) \cos (3 x)$

Step 2.4

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

$- \sin (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} 3 x) + lim_{x \to 0} 3 \cos (x) \cos (3 x)$

Step 2.5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$- \sin (lim_{x \to 0} x) \cdot \sin (3 lim_{x \to 0} x) + lim_{x \to 0} 3 \cos (x) \cos (3 x)$

Step 2.6

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$- \sin (lim_{x \to 0} x) \cdot \sin (3 lim_{x \to 0} x) + 3 lim_{x \to 0} \cos (x) \cos (3 x)$

Step 2.7

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

$- \sin (lim_{x \to 0} x) \cdot \sin (3 lim_{x \to 0} x) + 3 lim_{x \to 0} \cos (x) \cdot lim_{x \to 0} \cos (3 x)$

Step 2.8

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

$- \sin (lim_{x \to 0} x) \cdot \sin (3 lim_{x \to 0} x) + 3 \cos (lim_{x \to 0} x) \cdot lim_{x \to 0} \cos (3 x)$

Step 2.9

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

$- \sin (lim_{x \to 0} x) \cdot \sin (3 lim_{x \to 0} x) + 3 \cos (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} 3 x)$

Step 2.10

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$- \sin (lim_{x \to 0} x) \cdot \sin (3 lim_{x \to 0} x) + 3 \cos (lim_{x \to 0} x) \cdot \cos (3 lim_{x \to 0} x)$

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 $x$ .

$- \sin (0) \cdot \sin (3 lim_{x \to 0} x) + 3 \cos (lim_{x \to 0} x) \cdot \cos (3 lim_{x \to 0} x)$

Step 3.2

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

$- \sin (0) \cdot \sin (3 \cdot 0) + 3 \cos (lim_{x \to 0} x) \cdot \cos (3 lim_{x \to 0} x)$

Step 3.3

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

$- \sin (0) \cdot \sin (3 \cdot 0) + 3 \cos (0) \cdot \cos (3 lim_{x \to 0} x)$

Step 3.4

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

$- \sin (0) \cdot \sin (3 \cdot 0) + 3 \cos (0) \cdot \cos (3 \cdot 0)$

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 $\sin (0)$ is $0$ .

$- 0 \cdot \sin (3 \cdot 0) + 3 \cos (0) \cdot \cos (3 \cdot 0)$

Step 4.1.2

Multiply $- 1$ by $0$ .

$0 \cdot \sin (3 \cdot 0) + 3 \cos (0) \cdot \cos (3 \cdot 0)$

Step 4.1.3

Multiply $3$ by $0$ .

$0 \cdot \sin (0) + 3 \cos (0) \cdot \cos (3 \cdot 0)$

Step 4.1.4

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

$0 \cdot 0 + 3 \cos (0) \cdot \cos (3 \cdot 0)$

Step 4.1.5

Multiply $0$ by $0$ .

$0 + 3 \cos (0) \cdot \cos (3 \cdot 0)$

Step 4.1.6

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

$0 + 3 \cdot 1 \cdot \cos (3 \cdot 0)$

Step 4.1.7

Multiply $3$ by $1$ .

$0 + 3 \cdot \cos (3 \cdot 0)$

Step 4.1.8

Multiply $3$ by $0$ .

$0 + 3 \cdot \cos (0)$

Step 4.1.9

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

$0 + 3 \cdot 1$

Step 4.1.10

Multiply $3$ by $1$ .

$0 + 3$

Step 4.2

Add $0$ and $3$ .

$3$