Calculus Examples

lim x \to 0 \frac{sin (3 x)}{x}

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

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)}{lim x \to 0 x}

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

\frac{sin (lim x \to 0 3 x)}{lim x \to 0 x}

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

Step 1.1.2.1.2

Move the term

3

$3$ outside of the limit because it is constant with respect to

x

$x$ .

\frac{sin (3 lim x \to 0 x)}{lim x \to 0 x}

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

\frac{sin (3 lim x \to 0 x)}{lim x \to 0 x}

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

Step 1.1.2.2

Evaluate the limit of

x

$x$ by plugging in

0

$0$ for

x

$x$ .

\frac{sin (3 \cdot 0)}{lim x \to 0 x}

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

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Multiply

3

$3$ by

0

$0$ .

\frac{sin (0)}{lim x \to 0 x}

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

Step 1.1.2.3.2

The exact value of

sin (0)

$\sin (0)$ is

0

$0$ .

\frac{0}{lim x \to 0 x}

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

\frac{0}{lim x \to 0 x}

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

\frac{0}{lim x \to 0 x}

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

Step 1.1.3

Evaluate the limit of

x

$x$ by plugging in

0

$0$ for

x

$x$ .

\frac{0}{0}

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by

0

$0$ . The expression is undefined.

Undefined

\frac{0}{0}

$\frac{0}{0}$

Step 1.2

Since

\frac{0}{0}

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

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

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

$f (x) = \sin (x)$ and

$g (x) = 3 x$ .

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Step 1.3.2.1

To apply the Chain Rule, set

$u$ as

$3 x$ .

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

Step 1.3.2.2

The derivative of

$\sin (u)$ with respect to

$u$ is

$\cos (u)$ .

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

Step 1.3.2.3

Replace all occurrences of

$u$ with

$3 x$ .

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

Step 1.3.3

Since

$3$ is constant with respect to

$x$ , the derivative of

$3 x$ with respect to

$x$ is

$3 \frac{d}{d x} [x]$ .

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

Step 1.3.4

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.3.5

Multiply

$3$ by

$1$ .

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

Step 1.3.6

Move

$3$ to the left of

$\cos (3 x)$ .

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

Step 1.3.7

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.4

Divide

$3 \cos (3 x)$ by

$1$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Move the term

$3$ outside of the limit because it is constant with respect to

$x$ .

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

Step 2.2

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

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

Step 2.3

Move the term

$3$ outside of the limit because it is constant with respect to

$x$ .

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

Step 3

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

$3 \cos (3 \cdot 0)$

Step 4

Simplify the answer.

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Step 4.1

Multiply

$3$ by

$0$ .

$3 \cos (0)$

Step 4.2

The exact value of

$\cos (0)$ is

$1$ .

$3 \cdot 1$

Step 4.3

Multiply

$3$ by

$1$ .

$3$