Calculus Examples

$lim_{x \to 0} \frac{\sin (6 x)}{\sin (3 x)}$

Step 1

Multiply the numerator and denominator by $3 x$ .

$lim_{x \to 0} \frac{\sin (6 x) \cdot (3 x)}{\sin (3 x) \cdot (3 x)}$

Step 2

Multiply the numerator and denominator by $6 x$ .

$lim_{x \to 0} \frac{\sin (6 x) \cdot (3 x) \cdot (6 x)}{6 x \cdot \sin (3 x) \cdot (3 x)}$

Step 3

Separate fractions.

$lim_{x \to 0} \frac{\sin (6 x)}{6 x} \cdot \frac{3 x}{\sin (3 x)} \cdot \frac{6 x}{3 x}$

Step 4

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

$lim_{x \to 0} \frac{\sin (6 x)}{6 x} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5

The limit of $\frac{\sin (6 x)}{6 x}$ as $x$ approaches $0$ is $1$ .

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Step 5.1

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

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Step 5.1.1

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

$\frac{lim_{x \to 0} \sin (6 x)}{lim_{x \to 0} 6 x} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.1.2

Evaluate the limit of the numerator.

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Step 5.1.2.1

Evaluate the limit.

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Step 5.1.2.1.1

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

$\frac{\sin (lim_{x \to 0} 6 x)}{lim_{x \to 0} 6 x} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.1.2.1.2

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (6 lim_{x \to 0} x)}{lim_{x \to 0} 6 x} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.1.2.2

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

$\frac{\sin (6 \cdot 0)}{lim_{x \to 0} 6 x} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.1.2.3

Simplify the answer.

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Step 5.1.2.3.1

Multiply $6$ by $0$ .

$\frac{\sin (0)}{lim_{x \to 0} 6 x} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.1.2.3.2

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

$\frac{0}{lim_{x \to 0} 6 x} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.1.3

Evaluate the limit of the denominator.

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Step 5.1.3.1

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{6 lim_{x \to 0} x} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.1.3.2

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

$\frac{0}{6 \cdot 0} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.1.3.3

Multiply $6$ by $0$ .

$\frac{0}{0} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.1.3.4

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

Undefined

$\frac{0}{0} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.1.4

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

Undefined

$\frac{0}{0} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.2

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.

$lim_{x \to 0} \frac{\sin (6 x)}{6 x} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (6 x)]}{\frac{d}{d x} [6 x]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\sin (6 x)]}{\frac{d}{d x} [6 x]} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.3.2

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 $g (x) = 6 x$ .

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Step 5.3.2.1

To apply the Chain Rule, set $u$ as $6 x$ .

$lim_{x \to 0} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [6 x]}{\frac{d}{d x} [6 x]} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.3.2.2

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

$lim_{x \to 0} \frac{\cos (u) \frac{d}{d x} [6 x]}{\frac{d}{d x} [6 x]} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.3.2.3

Replace all occurrences of $u$ with $6 x$ .

$lim_{x \to 0} \frac{\cos (6 x) \frac{d}{d x} [6 x]}{\frac{d}{d x} [6 x]} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.3.3

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\cos (6 x) (6 \frac{d}{d x} [x])}{\frac{d}{d x} [6 x]} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.3.4

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

$lim_{x \to 0} \frac{\cos (6 x) (6 \cdot 1)}{\frac{d}{d x} [6 x]} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.3.5

Multiply $6$ by $1$ .

$lim_{x \to 0} \frac{\cos (6 x) \cdot 6}{\frac{d}{d x} [6 x]} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.3.6

Move $6$ to the left of $\cos (6 x)$ .

$lim_{x \to 0} \frac{6 \cdot \cos (6 x)}{\frac{d}{d x} [6 x]} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.3.7

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{6 \cos (6 x)}{6 \frac{d}{d x} [x]} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.3.8

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

$lim_{x \to 0} \frac{6 \cos (6 x)}{6 \cdot 1} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.3.9

Multiply $6$ by $1$ .

$lim_{x \to 0} \frac{6 \cos (6 x)}{6} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.4

Evaluate the limit.

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Step 5.4.1

Cancel the common factor of $6$ .

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Step 5.4.1.1

Cancel the common factor.

$lim_{x \to 0} \frac{6 \cos (6 x)}{6} \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.4.1.2

Divide $\cos (6 x)$ by $1$ .

$lim_{x \to 0} \cos (6 x) \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.4.2

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

$\cos (lim_{x \to 0} 6 x) \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.4.3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\cos (6 lim_{x \to 0} x) \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.5

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

$\cos (6 \cdot 0) \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.6

Simplify the answer.

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Step 5.6.1

Multiply $6$ by $0$ .

$\cos (0) \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 5.6.2

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

$1 \cdot lim_{x \to 0} \frac{3 x}{\sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6

The limit of $\frac{3 x}{\sin (3 x)}$ as $x$ approaches $0$ is $1$ .

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Step 6.1

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

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Step 6.1.1

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

$1 \cdot \frac{lim_{x \to 0} 3 x}{lim_{x \to 0} \sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.1.2

Evaluate the limit of the numerator.

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Step 6.1.2.1

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

$1 \cdot \frac{3 lim_{x \to 0} x}{lim_{x \to 0} \sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.1.2.2

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

$1 \cdot \frac{3 \cdot 0}{lim_{x \to 0} \sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.1.2.3

Multiply $3$ by $0$ .

$1 \cdot \frac{0}{lim_{x \to 0} \sin (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.1.3

Evaluate the limit of the denominator.

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Step 6.1.3.1

Evaluate the limit.

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Step 6.1.3.1.1

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

$1 \cdot \frac{0}{\sin (lim_{x \to 0} 3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.1.3.1.2

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

$1 \cdot \frac{0}{\sin (3 lim_{x \to 0} x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.1.3.2

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

$1 \cdot \frac{0}{\sin (3 \cdot 0)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.1.3.3

Simplify the answer.

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Step 6.1.3.3.1

Multiply $3$ by $0$ .

$1 \cdot \frac{0}{\sin (0)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.1.3.3.2

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

$1 \cdot \frac{0}{0} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.1.3.3.3

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

Undefined

$1 \cdot \frac{0}{0} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.1.3.4

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

Undefined

$1 \cdot \frac{0}{0} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.1.4

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

Undefined

$1 \cdot \frac{0}{0} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.2

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.

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

Step 6.3

Find the derivative of the numerator and denominator.

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Step 6.3.1

Differentiate the numerator and denominator.

$1 \cdot lim_{x \to 0} \frac{\frac{d}{d x} [3 x]}{\frac{d}{d x} [\sin (3 x)]} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.3.2

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

$1 \cdot lim_{x \to 0} \frac{3 \frac{d}{d x} [x]}{\frac{d}{d x} [\sin (3 x)]} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.3.3

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

$1 \cdot lim_{x \to 0} \frac{3 \cdot 1}{\frac{d}{d x} [\sin (3 x)]} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.3.4

Multiply $3$ by $1$ .

$1 \cdot lim_{x \to 0} \frac{3}{\frac{d}{d x} [\sin (3 x)]} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.3.5

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 6.3.5.1

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

$1 \cdot lim_{x \to 0} \frac{3}{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [3 x]} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.3.5.2

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

$1 \cdot lim_{x \to 0} \frac{3}{\cos (u) \frac{d}{d x} [3 x]} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.3.5.3

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

$1 \cdot lim_{x \to 0} \frac{3}{\cos (3 x) \frac{d}{d x} [3 x]} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.3.6

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

$1 \cdot lim_{x \to 0} \frac{3}{\cos (3 x) (3 \frac{d}{d x} [x])} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.3.7

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

$1 \cdot lim_{x \to 0} \frac{3}{\cos (3 x) (3 \cdot 1)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.3.8

Multiply $3$ by $1$ .

$1 \cdot lim_{x \to 0} \frac{3}{\cos (3 x) \cdot 3} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.3.9

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

$1 \cdot lim_{x \to 0} \frac{3}{3 \cos (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.4

Cancel the common factor of $3$ .

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Step 6.4.1

Cancel the common factor.

$1 \cdot lim_{x \to 0} \frac{3}{3 \cos (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.4.2

Rewrite the expression.

$1 \cdot lim_{x \to 0} \frac{1}{\cos (3 x)} \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.5

Convert from $\frac{1}{\cos (3 x)}$ to $\sec (3 x)$ .

$1 \cdot lim_{x \to 0} \sec (3 x) \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.6

Evaluate the limit.

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Step 6.6.1

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

$1 \cdot \sec (lim_{x \to 0} 3 x) \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.6.2

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

$1 \cdot \sec (3 lim_{x \to 0} x) \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.7

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

$1 \cdot \sec (3 \cdot 0) \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.8

Simplify the answer.

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Step 6.8.1

Multiply $3$ by $0$ .

$1 \cdot \sec (0) \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 6.8.2

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

$1 \cdot 1 \cdot lim_{x \to 0} \frac{6 x}{3 x}$

Step 7

Reduce.

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Step 7.1

Cancel the common factor of $6$ and $3$ .

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Step 7.1.1

Factor $3$ out of $6 x$ .

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

Step 7.1.2

Cancel the common factors.

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Step 7.1.2.1

Factor $3$ out of $3 x$ .

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

Step 7.1.2.2

Cancel the common factor.

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

Step 7.1.2.3

Rewrite the expression.

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

Step 7.2

Cancel the common factor of $x$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Divide $2$ by $1$ .

$1 \cdot 1 \cdot lim_{x \to 0} 2$

Step 8

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$1 \cdot 1 \cdot 2$

Step 9

Multiply $1 \cdot 1 \cdot 2$ .

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Step 9.1

Multiply $1$ by $1$ .

$1 \cdot 2$

Step 9.2

Multiply $2$ by $1$ .

$2$