Calculus Examples

$lim_{x \to 0} \frac{\sin^{2} (4 x)}{3 x}$

Step 1

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sin^{2} (4 x)}{x}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 2.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{1}{3} \cdot \frac{lim_{x \to 0} \sin^{2} (4 x)}{lim_{x \to 0} x}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

Move the exponent $2$ from $\sin^{2} (4 x)$ outside the limit using the Limits Power Rule.

$\frac{1}{3} \cdot \frac{{(lim_{x \to 0} \sin (4 x))}^{2}}{lim_{x \to 0} x}$

Step 2.1.2.1.2

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

$\frac{1}{3} \cdot \frac{\sin^{2} (lim_{x \to 0} 4 x)}{lim_{x \to 0} x}$

Step 2.1.2.1.3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} \cdot \frac{\sin^{2} (4 lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 2.1.2.2

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

$\frac{1}{3} \cdot \frac{\sin^{2} (4 \cdot 0)}{lim_{x \to 0} x}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Multiply $4$ by $0$ .

$\frac{1}{3} \cdot \frac{\sin^{2} (0)}{lim_{x \to 0} x}$

Step 2.1.2.3.2

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

$\frac{1}{3} \cdot \frac{0^{2}}{lim_{x \to 0} x}$

Step 2.1.2.3.3

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

$\frac{1}{3} \cdot \frac{0}{lim_{x \to 0} x}$

Step 2.1.3

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

$\frac{1}{3} \cdot \frac{0}{0}$

Step 2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{1}{3} \cdot \frac{0}{0}$

Step 2.2

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.

$lim_{x \to 0} \frac{\sin^{2} (4 x)}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin^{2} (4 x)]}{\frac{d}{d x} [x]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$\frac{1}{3} lim_{x \to 0} \frac{\frac{d}{d x} [\sin^{2} (4 x)]}{\frac{d}{d x} [x]}$

Step 2.3.2

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 $g (x) = \sin (4 x)$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u_{1}$ as $\sin (4 x)$ .

$\frac{1}{3} lim_{x \to 0} \frac{\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [x]}$

Step 2.3.2.2

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

$\frac{1}{3} lim_{x \to 0} \frac{2 u_{1} \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [x]}$

Step 2.3.2.3

Replace all occurrences of $u_{1}$ with $\sin (4 x)$ .

$\frac{1}{3} lim_{x \to 0} \frac{2 \sin (4 x) \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [x]}$

Step 2.3.3

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 $g (x) = 4 x$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u_{2}$ as $4 x$ .

$\frac{1}{3} lim_{x \to 0} \frac{2 \sin (4 x) \frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [4 x]}{\frac{d}{d x} [x]}$

Step 2.3.3.2

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

$\frac{1}{3} lim_{x \to 0} \frac{2 \sin (4 x) \cos (u_{2}) \frac{d}{d x} [4 x]}{\frac{d}{d x} [x]}$

Step 2.3.3.3

Replace all occurrences of $u_{2}$ with $4 x$ .

$\frac{1}{3} lim_{x \to 0} \frac{2 \sin (4 x) \cos (4 x) \frac{d}{d x} [4 x]}{\frac{d}{d x} [x]}$

Step 2.3.4

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{1}{3} lim_{x \to 0} \frac{2 \sin (4 x) \cos (4 x) \cdot 4 \frac{d}{d x} [x]}{\frac{d}{d x} [x]}$

Step 2.3.5

Multiply $4$ by $2$ .

$\frac{1}{3} lim_{x \to 0} \frac{8 \sin (4 x) \cos (4 x) \frac{d}{d x} [x]}{\frac{d}{d x} [x]}$

Step 2.3.6

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

$\frac{1}{3} lim_{x \to 0} \frac{8 \sin (4 x) \cos (4 x) \cdot 1}{\frac{d}{d x} [x]}$

Step 2.3.7

Multiply $8$ by $1$ .

$\frac{1}{3} lim_{x \to 0} \frac{8 \sin (4 x) \cos (4 x)}{\frac{d}{d x} [x]}$

Step 2.3.8

Reorder the factors of $8 \sin (4 x) \cos (4 x)$ .

$\frac{1}{3} lim_{x \to 0} \frac{8 \cos (4 x) \sin (4 x)}{\frac{d}{d x} [x]}$

Step 2.3.9

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

$\frac{1}{3} lim_{x \to 0} \frac{8 \cos (4 x) \sin (4 x)}{1}$

Step 2.4

Divide $8 \cos (4 x) \sin (4 x)$ by $1$ .

$\frac{1}{3} lim_{x \to 0} 8 \cos (4 x) \sin (4 x)$

Step 3

Evaluate the limit.

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Step 3.1

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} \cdot 8 lim_{x \to 0} \cos (4 x) \sin (4 x)$

Step 3.2

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

$\frac{1}{3} \cdot 8 lim_{x \to 0} \cos (4 x) \cdot lim_{x \to 0} \sin (4 x)$

Step 3.3

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

$\frac{1}{3} \cdot 8 \cos (lim_{x \to 0} 4 x) \cdot lim_{x \to 0} \sin (4 x)$

Step 3.4

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} \cdot 8 \cos (4 lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (4 x)$

Step 3.5

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

$\frac{1}{3} \cdot 8 \cos (4 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} 4 x)$

Step 3.6

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} \cdot 8 \cos (4 lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x)$

Step 4

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

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Step 4.1

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

$\frac{1}{3} \cdot 8 \cos (4 \cdot 0) \cdot \sin (4 lim_{x \to 0} x)$

Step 4.2

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

$\frac{1}{3} \cdot 8 \cos (4 \cdot 0) \cdot \sin (4 \cdot 0)$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{3}$ and $8$ .

$\frac{8}{3} \cos (4 \cdot 0) \cdot \sin (4 \cdot 0)$

Step 5.2

Multiply $4$ by $0$ .

$\frac{8}{3} \cos (0) \cdot \sin (4 \cdot 0)$

Step 5.3

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

$\frac{8}{3} \cdot 1 \cdot \sin (4 \cdot 0)$

Step 5.4

Multiply $\frac{8}{3}$ by $1$ .

$\frac{8}{3} \cdot \sin (4 \cdot 0)$

Step 5.5

Multiply $4$ by $0$ .

$\frac{8}{3} \cdot \sin (0)$

Step 5.6

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

$\frac{8}{3} \cdot 0$

Step 5.7

Multiply $\frac{8}{3}$ by $0$ .

$0$