Calculus Examples

$lim_{x \to 0} \frac{3 x^{2}}{1 - \cos (6 x)}$

Step 1

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

$3 lim_{x \to 0} \frac{x^{2}}{1 - \cos (6 x)}$

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.

$3 \frac{lim_{x \to 0} x^{2}}{lim_{x \to 0} 1 - \cos (6 x)}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$3 \frac{{(lim_{x \to 0} x)}^{2}}{lim_{x \to 0} 1 - \cos (6 x)}$

Step 2.1.2.2

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

$3 \frac{0^{2}}{lim_{x \to 0} 1 - \cos (6 x)}$

Step 2.1.2.3

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

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

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

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

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

Step 2.1.3.1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

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

Step 2.1.3.1.3

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

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

Step 2.1.3.1.4

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

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

Step 2.1.3.2

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

$3 \frac{0}{1 - \cos (6 \cdot 0)}$

Step 2.1.3.3

Simplify the answer.

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Step 2.1.3.3.1

Simplify each term.

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Step 2.1.3.3.1.1

Multiply $6$ by $0$ .

$3 \frac{0}{1 - \cos (0)}$

Step 2.1.3.3.1.2

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

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

Step 2.1.3.3.1.3

Multiply $- 1$ by $1$ .

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

Step 2.1.3.3.2

Subtract $1$ from $1$ .

$3 (\frac{0}{0})$

Step 2.1.3.3.3

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

Undefined

$3 (\frac{0}{0})$

Step 2.1.3.4

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

Undefined

$3 (\frac{0}{0})$

Step 2.1.4

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

Undefined

$3 (\frac{0}{0})$

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.

$lim_{x \to 0} \frac{x^{2}}{1 - \cos (6 x)} = lim_{x \to 0} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [1 - \cos (6 x)]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$3 lim_{x \to 0} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [1 - \cos (6 x)]}$

Step 2.3.2

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

$3 lim_{x \to 0} \frac{2 x}{\frac{d}{d x} [1 - \cos (6 x)]}$

Step 2.3.3

By the Sum Rule, the derivative of $1 - \cos (6 x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (6 x)]$ .

$3 lim_{x \to 0} \frac{2 x}{\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (6 x)]}$

Step 2.3.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$3 lim_{x \to 0} \frac{2 x}{0 + \frac{d}{d x} [- \cos (6 x)]}$

Step 2.3.5

Evaluate $\frac{d}{d x} [- \cos (6 x)]$ .

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Step 2.3.5.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (6 x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (6 x)]$ .

$3 lim_{x \to 0} \frac{2 x}{0 - \frac{d}{d x} [\cos (6 x)]}$

Step 2.3.5.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 6 x$ .

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Step 2.3.5.2.1

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

$3 lim_{x \to 0} \frac{2 x}{0 - \frac{d}{d u} [\cos (u)] \frac{d}{d x} [6 x]}$

Step 2.3.5.2.2

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

$3 lim_{x \to 0} \frac{2 x}{0 - - 1 \sin (u) \frac{d}{d x} [6 x]}$

Step 2.3.5.2.3

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

$3 lim_{x \to 0} \frac{2 x}{0 - - 1 \sin (6 x) \frac{d}{d x} [6 x]}$

Step 2.3.5.3

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

$3 lim_{x \to 0} \frac{2 x}{0 - - 1 \sin (6 x) \cdot 6 \frac{d}{d x} [x]}$

Step 2.3.5.4

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

$3 lim_{x \to 0} \frac{2 x}{0 - - 1 \sin (6 x) \cdot 6 \cdot 1}$

Step 2.3.5.5

Multiply $6$ by $1$ .

$3 lim_{x \to 0} \frac{2 x}{0 - - 1 \sin (6 x) \cdot 6}$

Step 2.3.5.6

Multiply $6$ by $- 1$ .

$3 lim_{x \to 0} \frac{2 x}{0 - - 6 \sin (6 x)}$

Step 2.3.5.7

Multiply $- 6$ by $- 1$ .

$3 lim_{x \to 0} \frac{2 x}{0 + 6 \sin (6 x)}$

Step 2.3.6

Add $0$ and $6 \sin (6 x)$ .

$3 lim_{x \to 0} \frac{2 x}{6 \sin (6 x)}$

Step 2.4

Cancel the common factor of $2$ and $6$ .

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Step 2.4.1

Factor $2$ out of $2 x$ .

$3 lim_{x \to 0} \frac{2 (x)}{6 \sin (6 x)}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $2$ out of $6 \sin (6 x)$ .

$3 lim_{x \to 0} \frac{2 (x)}{2 \cdot 3 \sin (6 x)}$

Step 2.4.2.2

Cancel the common factor.

$3 lim_{x \to 0} \frac{2 x}{2 \cdot 3 \sin (6 x)}$

Step 2.4.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

Apply L'Hospital's rule.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

Evaluate the limit of the denominator.

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Step 4.1.3.1

Evaluate the limit.

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Step 4.1.3.1.1

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

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

Step 4.1.3.1.2

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

Simplify the answer.

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Step 4.1.3.3.1

Multiply $6$ by $0$ .

$3 (\frac{1}{3}) \frac{0}{\sin (0)}$

Step 4.1.3.3.2

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

$3 (\frac{1}{3}) \frac{0}{0}$

Step 4.1.3.3.3

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

Undefined

$3 (\frac{1}{3}) \frac{0}{0}$

Step 4.1.3.4

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

Undefined

$3 (\frac{1}{3}) \frac{0}{0}$

Step 4.1.4

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

Undefined

$3 (\frac{1}{3}) \frac{0}{0}$

Step 4.2

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.

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

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

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

Step 4.3.2

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

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

Step 4.3.3

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 $g (x) = 6 x$ .

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Step 4.3.3.1

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

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

Step 4.3.3.2

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

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

Step 4.3.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

Multiply $6$ by $1$ .

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

Step 4.3.7

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

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

Step 5

Evaluate the limit.

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Step 5.1

Move the term $\frac{1}{6}$ outside of the limit because it is constant with respect to $x$ .

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

Step 5.2

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

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

Step 5.3

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

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

Step 5.4

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

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

Step 5.5

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

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

Cancel the common factor of $3$ .

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Step 7.1.1

Factor $3$ out of $6$ .

$3 (\frac{1}{3}) \frac{1}{3 (2)} \frac{1}{\cos (6 \cdot 0)}$

Step 7.1.2

Cancel the common factor.

$3 (\frac{1}{3}) \frac{1}{3 \cdot 2} \frac{1}{\cos (6 \cdot 0)}$

Step 7.1.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{1}{2} \frac{1}{\cos (6 \cdot 0)}$

Step 7.2

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

$\frac{1}{3 \cdot 2} \cdot \frac{1}{\cos (6 \cdot 0)}$

Step 7.3

Multiply $3$ by $2$ .

$\frac{1}{6} \cdot \frac{1}{\cos (6 \cdot 0)}$

Step 7.4

Convert from $\frac{1}{\cos (6 \cdot 0)}$ to $\sec (6 \cdot 0)$ .

$\frac{1}{6} \sec (6 \cdot 0)$

Step 7.5

Multiply $6$ by $0$ .

$\frac{1}{6} \sec (0)$

Step 7.6

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

$\frac{1}{6} \cdot 1$

Step 7.7

Multiply $\frac{1}{6}$ by $1$ .

$\frac{1}{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{6}$

Decimal Form:

$0.1\overline{6}$