Calculus Examples

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

Step 1

Move the term $\frac{3}{4}$ outside of the limit because it is constant with respect to $x$ .

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

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.

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

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 limit inside the trig function because sine is continuous.

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

Step 2.1.2.1.2

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Multiply $3$ by $0$ .

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

Step 2.1.2.3.2

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

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

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Evaluate the limit.

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Step 2.1.3.1.1

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

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

Step 2.1.3.1.2

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

Simplify the answer.

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Step 2.1.3.3.1

Multiply $4$ by $0$ .

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

Step 2.1.3.3.2

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

$\frac{3}{4} \cdot \frac{0}{0}$

Step 2.1.3.3.3

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

Undefined

$\frac{3}{4} \cdot \frac{0}{0}$

Step 2.1.3.4

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

Undefined

$\frac{3}{4} \cdot \frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{3}{4} \cdot \frac{0}{0}$

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

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

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Multiply $3$ by $1$ .

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

Step 2.3.6

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

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

Step 2.3.7

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 2.3.7.1

To apply the Chain Rule, set $u$ as $4 x$ .

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

Step 2.3.7.2

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

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

Step 2.3.7.3

Replace all occurrences of $u$ with $4 x$ .

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

Step 2.3.8

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

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

Step 2.3.9

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

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

Step 2.3.10

Multiply $4$ by $1$ .

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

Step 2.3.11

Move $4$ to the left of $\cos (4 x)$ .

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

Step 3

Evaluate the limit.

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Step 3.1

Move the term $\frac{3}{4}$ outside of the limit because it is constant with respect to $x$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

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 $x$ .

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

Step 4.2

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

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

Step 5

Simplify the answer.

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Step 5.1

Multiply $\frac{3}{4} \cdot \frac{3}{4}$ .

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Step 5.1.1

Multiply $\frac{3}{4}$ by $\frac{3}{4}$ .

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

Step 5.1.2

Multiply $3$ by $3$ .

$\frac{9}{4 \cdot 4} \cdot \frac{\cos (3 \cdot 0)}{\cos (4 \cdot 0)}$

Step 5.1.3

Multiply $4$ by $4$ .

$\frac{9}{16} \cdot \frac{\cos (3 \cdot 0)}{\cos (4 \cdot 0)}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Multiply $3$ by $0$ .

$\frac{9}{16} \cdot \frac{\cos (0)}{\cos (4 \cdot 0)}$

Step 5.2.2

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

$\frac{9}{16} \cdot \frac{1}{\cos (4 \cdot 0)}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Multiply $4$ by $0$ .

$\frac{9}{16} \cdot \frac{1}{\cos (0)}$

Step 5.3.2

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

$\frac{9}{16} \cdot \frac{1}{1}$

Step 5.4

Cancel the common factor of $1$ .

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Step 5.4.1

Cancel the common factor.

$\frac{9}{16} \cdot \frac{1}{1}$

Step 5.4.2

Rewrite the expression.

$\frac{9}{16} \cdot 1$

Step 5.5

Multiply $\frac{9}{16}$ by $1$ .

$\frac{9}{16}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{16}$

Decimal Form:

$0.5625$