Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Move the exponent $2$ from $\sec^{2} (3 x)$ outside the limit using the Limits Power Rule.

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

Step 1.2.1.4

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

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

Step 1.2.1.5

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

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

Step 1.2.2

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

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Reorder $1$ and $- \sec^{2} (3 \cdot 0)$ .

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

Step 1.2.3.2

Factor $- 1$ out of $- \sec^{2} (3 \cdot 0)$ .

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

Step 1.2.3.3

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 1.2.3.4

Factor $- 1$ out of $- \sec^{2} (3 \cdot 0) - 1 \cdot - 1$ .

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

Step 1.2.3.5

Apply pythagorean identity.

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

Step 1.2.3.6

Multiply $3$ by $0$ .

$\frac{- \tan^{2} (0)}{lim_{x \to 0} {(6 x)}^{2}}$

Step 1.2.3.7

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

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

Step 1.2.3.8

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

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

Step 1.2.3.9

Multiply $- 1$ by $0$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

Move the exponent $2$ from ${(6 x)}^{2}$ outside the limit using the Limits Power Rule.

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

Step 1.3.1.2

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

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

Step 1.3.2

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

$\frac{0}{{(6 \cdot 0)}^{2}}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Multiply $6$ by $0$ .

$\frac{0}{0^{2}}$

Step 1.3.3.2

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

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

By the Sum Rule, the derivative of $1 - \sec^{2} (3 x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \sec^{2} (3 x)]$ .

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

Step 3.3

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

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

Step 3.4

Evaluate $\frac{d}{d x} [- \sec^{2} (3 x)]$ .

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Step 3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sec^{2} (3 x)$ with respect to $x$ is $- \frac{d}{d x} [\sec^{2} (3 x)]$ .

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

Step 3.4.2

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 $g (x) = \sec (3 x)$ .

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Step 3.4.2.1

To apply the Chain Rule, set $u_{1}$ as $\sec (3 x)$ .

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

Step 3.4.2.2

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

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

Step 3.4.2.3

Replace all occurrences of $u_{1}$ with $\sec (3 x)$ .

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

Step 3.4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 3 x$ .

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Step 3.4.3.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

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

Step 3.4.3.2

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

$lim_{x \to 0} \frac{0 - (2 \sec (3 x) (\sec (u_{2}) \tan (u_{2}) \frac{d}{d x} [3 x]))}{\frac{d}{d x} [{(6 x)}^{2}]}$

Step 3.4.3.3

Replace all occurrences of $u_{2}$ with $3 x$ .

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

Step 3.4.4

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

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

Step 3.4.5

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

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

Step 3.4.6

Multiply $3$ by $1$ .

$lim_{x \to 0} \frac{0 - (2 \sec (3 x) (\sec (3 x) \tan (3 x) \cdot 3))}{\frac{d}{d x} [{(6 x)}^{2}]}$

Step 3.4.7

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

$lim_{x \to 0} \frac{0 - (2 \sec (3 x) (3 \cdot (\sec (3 x) \tan (3 x))))}{\frac{d}{d x} [{(6 x)}^{2}]}$

Step 3.4.8

Remove parentheses.

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

Step 3.4.9

Multiply $3$ by $2$ .

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

Step 3.4.10

Raise $\sec (3 x)$ to the power of $1$ .

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

Step 3.4.11

Raise $\sec (3 x)$ to the power of $1$ .

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

Step 3.4.12

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.4.13

Add $1$ and $1$ .

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

Step 3.4.14

Multiply $6$ by $- 1$ .

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

Step 3.4.15

Remove parentheses.

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

Step 3.5

Simplify.

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Step 3.5.1

Subtract $6 \sec^{2} (3 x) \tan (3 x)$ from $0$ .

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

Step 3.5.2

Rewrite $\sec (3 x)$ in terms of sines and cosines.

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

Step 3.5.3

Apply the product rule to $\frac{1}{\cos (3 x)}$ .

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

Step 3.5.4

One to any power is one.

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

Step 3.5.5

Combine $- 6$ and $\frac{1}{\cos^{2} (3 x)}$ .

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

Step 3.5.6

Move the negative in front of the fraction.

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

Step 3.5.7

Rewrite $\tan (3 x)$ in terms of sines and cosines.

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

Step 3.5.8

Multiply $- \frac{6}{\cos^{2} (3 x)} \cdot \frac{\sin (3 x)}{\cos (3 x)}$ .

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Step 3.5.8.1

Multiply $\frac{\sin (3 x)}{\cos (3 x)}$ by $\frac{6}{\cos^{2} (3 x)}$ .

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

Step 3.5.8.2

Multiply $\cos (3 x)$ by $\cos^{2} (3 x)$ by adding the exponents.

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Step 3.5.8.2.1

Multiply $\cos (3 x)$ by $\cos^{2} (3 x)$ .

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Step 3.5.8.2.1.1

Raise $\cos (3 x)$ to the power of $1$ .

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

Step 3.5.8.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5.8.2.2

Add $1$ and $2$ .

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

Step 3.5.9

Move $6$ to the left of $\sin (3 x)$ .

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

Step 3.6

Apply the product rule to $6 x$ .

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

Step 3.7

Raise $6$ to the power of $2$ .

$lim_{x \to 0} \frac{- \frac{6 \sin (3 x)}{\cos^{3} (3 x)}}{\frac{d}{d x} [36 x^{2}]}$

Step 3.8

Since $36$ is constant with respect to $x$ , the derivative of $36 x^{2}$ with respect to $x$ is $36 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 0} \frac{- \frac{6 \sin (3 x)}{\cos^{3} (3 x)}}{36 \frac{d}{d x} [x^{2}]}$

Step 3.9

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

$lim_{x \to 0} \frac{- \frac{6 \sin (3 x)}{\cos^{3} (3 x)}}{36 (2 x)}$

Step 3.10

Multiply $2$ by $36$ .

$lim_{x \to 0} \frac{- \frac{6 \sin (3 x)}{\cos^{3} (3 x)}}{72 x}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} - \frac{6 \sin (3 x)}{\cos^{3} (3 x)} \cdot \frac{1}{72 x}$

Step 5

Evaluate the limit.

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Step 5.1

Multiply $\frac{1}{72 x}$ by $\frac{6 \sin (3 x)}{\cos^{3} (3 x)}$ .

$lim_{x \to 0} - \frac{6 \sin (3 x)}{72 x \cos^{3} (3 x)}$

Step 5.2

Cancel the common factor of $6$ and $72$ .

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Step 5.2.1

Factor $6$ out of $6 \sin (3 x)$ .

$lim_{x \to 0} - \frac{6 (\sin (3 x))}{72 x \cos^{3} (3 x)}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $6$ out of $72 x \cos^{3} (3 x)$ .

$lim_{x \to 0} - \frac{6 (\sin (3 x))}{6 (12 x \cos^{3} (3 x))}$

Step 5.2.2.2

Cancel the common factor.

$lim_{x \to 0} - \frac{6 \sin (3 x)}{6 (12 x \cos^{3} (3 x))}$

Step 5.2.2.3

Rewrite the expression.

$lim_{x \to 0} - \frac{\sin (3 x)}{12 x \cos^{3} (3 x)}$

Step 5.3

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to 0} \frac{\sin (3 x)}{12 x \cos^{3} (3 x)}$

Step 5.4

Move the term $\frac{1}{12}$ outside of the limit because it is constant with respect to $x$ .

$- \frac{1}{12} lim_{x \to 0} \frac{\sin (3 x)}{x \cos^{3} (3 x)}$

Step 6

Apply L'Hospital's rule.

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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.

$- \frac{1}{12} \cdot \frac{lim_{x \to 0} \sin (3 x)}{lim_{x \to 0} x \cos^{3} (3 x)}$

Step 6.1.2

Evaluate the limit of the numerator.

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Step 6.1.2.1

Evaluate the limit.

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Step 6.1.2.1.1

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

$- \frac{1}{12} \cdot \frac{\sin (lim_{x \to 0} 3 x)}{lim_{x \to 0} x \cos^{3} (3 x)}$

Step 6.1.2.1.2

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

$- \frac{1}{12} \cdot \frac{\sin (3 lim_{x \to 0} x)}{lim_{x \to 0} x \cos^{3} (3 x)}$

Step 6.1.2.2

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

$- \frac{1}{12} \cdot \frac{\sin (3 \cdot 0)}{lim_{x \to 0} x \cos^{3} (3 x)}$

Step 6.1.2.3

Simplify the answer.

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Step 6.1.2.3.1

Multiply $3$ by $0$ .

$- \frac{1}{12} \cdot \frac{\sin (0)}{lim_{x \to 0} x \cos^{3} (3 x)}$

Step 6.1.2.3.2

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

$- \frac{1}{12} \cdot \frac{0}{lim_{x \to 0} x \cos^{3} (3 x)}$

Step 6.1.3

Evaluate the limit of the denominator.

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Step 6.1.3.1

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

$- \frac{1}{12} \cdot \frac{0}{lim_{x \to 0} x \cdot lim_{x \to 0} \cos^{3} (3 x)}$

Step 6.1.3.2

Move the exponent $3$ from $\cos^{3} (3 x)$ outside the limit using the Limits Power Rule.

$- \frac{1}{12} \cdot \frac{0}{lim_{x \to 0} x \cdot {(lim_{x \to 0} \cos (3 x))}^{3}}$

Step 6.1.3.3

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

$- \frac{1}{12} \cdot \frac{0}{lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} 3 x)}$

Step 6.1.3.4

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

$- \frac{1}{12} \cdot \frac{0}{lim_{x \to 0} x \cdot \cos^{3} (3 lim_{x \to 0} x)}$

Step 6.1.3.5

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

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Step 6.1.3.5.1

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

$- \frac{1}{12} \cdot \frac{0}{0 \cdot \cos^{3} (3 lim_{x \to 0} x)}$

Step 6.1.3.5.2

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

$- \frac{1}{12} \cdot \frac{0}{0 \cdot \cos^{3} (3 \cdot 0)}$

Step 6.1.3.6

Simplify the answer.

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Step 6.1.3.6.1

Multiply $3$ by $0$ .

$- \frac{1}{12} \cdot \frac{0}{0 \cdot \cos^{3} (0)}$

Step 6.1.3.6.2

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

$- \frac{1}{12} \cdot \frac{0}{0 \cdot 1^{3}}$

Step 6.1.3.6.3

One to any power is one.

$- \frac{1}{12} \cdot \frac{0}{0 \cdot 1}$

Step 6.1.3.6.4

Multiply $0$ by $1$ .

$- \frac{1}{12} \cdot \frac{0}{0}$

Step 6.1.3.6.5

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

Undefined

$- \frac{1}{12} \cdot \frac{0}{0}$

Step 6.1.3.7

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

Undefined

$- \frac{1}{12} \cdot \frac{0}{0}$

Step 6.1.4

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

Undefined

$- \frac{1}{12} \cdot \frac{0}{0}$

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.

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

Step 6.3

Find the derivative of the numerator and denominator.

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Step 6.3.1

Differentiate the numerator and denominator.

$- \frac{1}{12} lim_{x \to 0} \frac{\frac{d}{d x} [\sin (3 x)]}{\frac{d}{d x} [x \cos^{3} (3 x)]}$

Step 6.3.2

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 6.3.2.1

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

$- \frac{1}{12} lim_{x \to 0} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [3 x]}{\frac{d}{d x} [x \cos^{3} (3 x)]}$

Step 6.3.2.2

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

$- \frac{1}{12} lim_{x \to 0} \frac{\cos (u) \frac{d}{d x} [3 x]}{\frac{d}{d x} [x \cos^{3} (3 x)]}$

Step 6.3.2.3

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

$- \frac{1}{12} lim_{x \to 0} \frac{\cos (3 x) \frac{d}{d x} [3 x]}{\frac{d}{d x} [x \cos^{3} (3 x)]}$

Step 6.3.3

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

$- \frac{1}{12} lim_{x \to 0} \frac{\cos (3 x) \cdot 3 \frac{d}{d x} [x]}{\frac{d}{d x} [x \cos^{3} (3 x)]}$

Step 6.3.4

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

$- \frac{1}{12} lim_{x \to 0} \frac{\cos (3 x) \cdot 3 \cdot 1}{\frac{d}{d x} [x \cos^{3} (3 x)]}$

Step 6.3.5

Multiply $3$ by $1$ .

$- \frac{1}{12} lim_{x \to 0} \frac{\cos (3 x) \cdot 3}{\frac{d}{d x} [x \cos^{3} (3 x)]}$

Step 6.3.6

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

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cdot \cos (3 x)}{\frac{d}{d x} [x \cos^{3} (3 x)]}$

Step 6.3.7

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \cos^{3} (3 x)$ .

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{x \frac{d}{d x} [\cos^{3} (3 x)] + \cos^{3} (3 x) \frac{d}{d x} [x]}$

Step 6.3.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \cos (3 x)$ .

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Step 6.3.8.1

To apply the Chain Rule, set $u_{1}$ as $\cos (3 x)$ .

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{x \frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [\cos (3 x)] + \cos^{3} (3 x) \frac{d}{d x} [x]}$

Step 6.3.8.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{x \cdot 3 {u_{1}}^{2} \frac{d}{d x} [\cos (3 x)] + \cos^{3} (3 x) \frac{d}{d x} [x]}$

Step 6.3.8.3

Replace all occurrences of $u_{1}$ with $\cos (3 x)$ .

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{x \cdot 3 \cos^{2} (3 x) \frac{d}{d x} [\cos (3 x)] + \cos^{3} (3 x) \frac{d}{d x} [x]}$

Step 6.3.9

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 $g (x) = 3 x$ .

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Step 6.3.9.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{x \cdot 3 \cos^{2} (3 x) \frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [3 x] + \cos^{3} (3 x) \frac{d}{d x} [x]}$

Step 6.3.9.2

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

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{x \cdot 3 \cos^{2} (3 x) \cdot - 1 \sin (u_{2}) \frac{d}{d x} [3 x] + \cos^{3} (3 x) \frac{d}{d x} [x]}$

Step 6.3.9.3

Replace all occurrences of $u_{2}$ with $3 x$ .

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{x \cdot 3 \cos^{2} (3 x) \cdot - 1 \sin (3 x) \frac{d}{d x} [3 x] + \cos^{3} (3 x) \frac{d}{d x} [x]}$

Step 6.3.10

Multiply $- 1$ by $3$ .

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{x \cdot - 3 \cos^{2} (3 x) \sin (3 x) \frac{d}{d x} [3 x] + \cos^{3} (3 x) \frac{d}{d x} [x]}$

Step 6.3.11

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

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{x \cdot - 3 \cos^{2} (3 x) \sin (3 x) \cdot 3 \frac{d}{d x} [x] + \cos^{3} (3 x) \frac{d}{d x} [x]}$

Step 6.3.12

Multiply $3$ by $- 3$ .

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{x \cdot - 9 \cos^{2} (3 x) \sin (3 x) \frac{d}{d x} [x] + \cos^{3} (3 x) \frac{d}{d x} [x]}$

Step 6.3.13

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

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{x \cdot - 9 \cos^{2} (3 x) \sin (3 x) \cdot 1 + \cos^{3} (3 x) \frac{d}{d x} [x]}$

Step 6.3.14

Multiply $- 9$ by $1$ .

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{x \cdot - 9 \cos^{2} (3 x) \sin (3 x) + \cos^{3} (3 x) \frac{d}{d x} [x]}$

Step 6.3.15

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

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{x \cdot - 9 \cos^{2} (3 x) \sin (3 x) + \cos^{3} (3 x) \cdot 1}$

Step 6.3.16

Multiply $\cos^{3} (3 x)$ by $1$ .

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{x \cdot - 9 \cos^{2} (3 x) \sin (3 x) + \cos^{3} (3 x)}$

Step 6.3.17

Reorder terms.

$- \frac{1}{12} lim_{x \to 0} \frac{3 \cos (3 x)}{- 9 x \cos^{2} (3 x) \sin (3 x) + \cos^{3} (3 x)}$

Step 7

Evaluate the limit.

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Step 7.1

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

$- \frac{1}{12} \cdot 3 lim_{x \to 0} \frac{\cos (3 x)}{- 9 x \cos^{2} (3 x) \sin (3 x) + \cos^{3} (3 x)}$

Step 7.2

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

$- \frac{1}{12} \cdot 3 \frac{lim_{x \to 0} \cos (3 x)}{lim_{x \to 0} - 9 x \cos^{2} (3 x) \sin (3 x) + \cos^{3} (3 x)}$

Step 7.3

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

$- \frac{1}{12} \cdot 3 \frac{\cos (lim_{x \to 0} 3 x)}{lim_{x \to 0} - 9 x \cos^{2} (3 x) \sin (3 x) + \cos^{3} (3 x)}$

Step 7.4

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

$- \frac{1}{12} \cdot 3 \frac{\cos (3 lim_{x \to 0} x)}{lim_{x \to 0} - 9 x \cos^{2} (3 x) \sin (3 x) + \cos^{3} (3 x)}$

Step 7.5

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

$- \frac{1}{12} \cdot 3 \frac{\cos (3 lim_{x \to 0} x)}{- lim_{x \to 0} 9 x \cos^{2} (3 x) \sin (3 x) + lim_{x \to 0} \cos^{3} (3 x)}$

Step 7.6

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$- \frac{1}{12} \cdot 3 \frac{\cos (3 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cos^{2} (3 x) \sin (3 x) + lim_{x \to 0} \cos^{3} (3 x)}$

Step 7.7

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

$- \frac{1}{12} \cdot 3 \frac{\cos (3 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot lim_{x \to 0} \cos^{2} (3 x) \cdot lim_{x \to 0} \sin (3 x) + lim_{x \to 0} \cos^{3} (3 x)}$

Step 7.8

Move the exponent $2$ from $\cos^{2} (3 x)$ outside the limit using the Limits Power Rule.

$- \frac{1}{12} \cdot 3 \frac{\cos (3 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot {(lim_{x \to 0} \cos (3 x))}^{2} \cdot lim_{x \to 0} \sin (3 x) + lim_{x \to 0} \cos^{3} (3 x)}$

Step 7.9

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

$- \frac{1}{12} \cdot 3 \frac{\cos (3 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} 3 x) \cdot lim_{x \to 0} \sin (3 x) + lim_{x \to 0} \cos^{3} (3 x)}$

Step 7.10

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

$- \frac{1}{12} \cdot 3 \frac{\cos (3 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \cos^{2} (3 lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (3 x) + lim_{x \to 0} \cos^{3} (3 x)}$

Step 7.11

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

$- \frac{1}{12} \cdot 3 \frac{\cos (3 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \cos^{2} (3 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} 3 x) + lim_{x \to 0} \cos^{3} (3 x)}$

Step 7.12

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

$- \frac{1}{12} \cdot 3 \frac{\cos (3 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \cos^{2} (3 lim_{x \to 0} x) \cdot \sin (3 lim_{x \to 0} x) + lim_{x \to 0} \cos^{3} (3 x)}$

Step 7.13

Move the exponent $3$ from $\cos^{3} (3 x)$ outside the limit using the Limits Power Rule.

$- \frac{1}{12} \cdot 3 \frac{\cos (3 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \cos^{2} (3 lim_{x \to 0} x) \cdot \sin (3 lim_{x \to 0} x) + {(lim_{x \to 0} \cos (3 x))}^{3}}$

Step 7.14

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

$- \frac{1}{12} \cdot 3 \frac{\cos (3 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \cos^{2} (3 lim_{x \to 0} x) \cdot \sin (3 lim_{x \to 0} x) + \cos^{3} (lim_{x \to 0} 3 x)}$

Step 7.15

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

$- \frac{1}{12} \cdot 3 \frac{\cos (3 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \cos^{2} (3 lim_{x \to 0} x) \cdot \sin (3 lim_{x \to 0} x) + \cos^{3} (3 lim_{x \to 0} x)}$

Step 8

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

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Step 8.1

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

$- \frac{1}{12} \cdot 3 \frac{\cos (3 \cdot 0)}{- 9 lim_{x \to 0} x \cdot \cos^{2} (3 lim_{x \to 0} x) \cdot \sin (3 lim_{x \to 0} x) + \cos^{3} (3 lim_{x \to 0} x)}$

Step 8.2

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

$- \frac{1}{12} \cdot 3 \frac{\cos (3 \cdot 0)}{- 9 \cdot 0 \cdot \cos^{2} (3 lim_{x \to 0} x) \cdot \sin (3 lim_{x \to 0} x) + \cos^{3} (3 lim_{x \to 0} x)}$

Step 8.3

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

$- \frac{1}{12} \cdot 3 \frac{\cos (3 \cdot 0)}{- 9 \cdot 0 \cdot \cos^{2} (3 \cdot 0) \cdot \sin (3 lim_{x \to 0} x) + \cos^{3} (3 lim_{x \to 0} x)}$

Step 8.4

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

$- \frac{1}{12} \cdot 3 \frac{\cos (3 \cdot 0)}{- 9 \cdot 0 \cdot \cos^{2} (3 \cdot 0) \cdot \sin (3 \cdot 0) + \cos^{3} (3 lim_{x \to 0} x)}$

Step 8.5

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

$- \frac{1}{12} \cdot 3 \frac{\cos (3 \cdot 0)}{- 9 \cdot 0 \cdot \cos^{2} (3 \cdot 0) \cdot \sin (3 \cdot 0) + \cos^{3} (3 \cdot 0)}$

Step 9

Simplify the answer.

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Step 9.1

Cancel the common factor of $3$ .

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Step 9.1.1

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$\frac{- 1}{12} \cdot 3 \frac{\cos (3 \cdot 0)}{- 9 \cdot 0 \cdot \cos^{2} (3 \cdot 0) \cdot \sin (3 \cdot 0) + \cos^{3} (3 \cdot 0)}$

Step 9.1.2

Factor $3$ out of $12$ .

$\frac{- 1}{3 (4)} \cdot 3 \frac{\cos (3 \cdot 0)}{- 9 \cdot 0 \cdot \cos^{2} (3 \cdot 0) \cdot \sin (3 \cdot 0) + \cos^{3} (3 \cdot 0)}$

Step 9.1.3

Cancel the common factor.

$\frac{- 1}{3 \cdot 4} \cdot 3 \frac{\cos (3 \cdot 0)}{- 9 \cdot 0 \cdot \cos^{2} (3 \cdot 0) \cdot \sin (3 \cdot 0) + \cos^{3} (3 \cdot 0)}$

Step 9.1.4

Rewrite the expression.

$\frac{- 1}{4} \cdot \frac{\cos (3 \cdot 0)}{- 9 \cdot 0 \cdot \cos^{2} (3 \cdot 0) \cdot \sin (3 \cdot 0) + \cos^{3} (3 \cdot 0)}$

Step 9.2

Move the negative in front of the fraction.

$- \frac{1}{4} \cdot \frac{\cos (3 \cdot 0)}{- 9 \cdot 0 \cdot \cos^{2} (3 \cdot 0) \cdot \sin (3 \cdot 0) + \cos^{3} (3 \cdot 0)}$

Step 9.3

Simplify the numerator.

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Step 9.3.1

Multiply $3$ by $0$ .

$- \frac{1}{4} \cdot \frac{\cos (0)}{- 9 \cdot 0 \cdot \cos^{2} (3 \cdot 0) \cdot \sin (3 \cdot 0) + \cos^{3} (3 \cdot 0)}$

Step 9.3.2

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

$- \frac{1}{4} \cdot \frac{1}{- 9 \cdot 0 \cdot \cos^{2} (3 \cdot 0) \cdot \sin (3 \cdot 0) + \cos^{3} (3 \cdot 0)}$

Step 9.4

Simplify the denominator.

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Step 9.4.1

Multiply $- 9$ by $0$ .

$- \frac{1}{4} \cdot \frac{1}{0 \cdot \cos^{2} (3 \cdot 0) \cdot \sin (3 \cdot 0) + \cos^{3} (3 \cdot 0)}$

Step 9.4.2

Multiply $3$ by $0$ .

$- \frac{1}{4} \cdot \frac{1}{0 \cdot \cos^{2} (0) \cdot \sin (3 \cdot 0) + \cos^{3} (3 \cdot 0)}$

Step 9.4.3

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

$- \frac{1}{4} \cdot \frac{1}{0 \cdot 1^{2} \cdot \sin (3 \cdot 0) + \cos^{3} (3 \cdot 0)}$

Step 9.4.4

One to any power is one.

$- \frac{1}{4} \cdot \frac{1}{0 \cdot 1 \cdot \sin (3 \cdot 0) + \cos^{3} (3 \cdot 0)}$

Step 9.4.5

Multiply $0$ by $1$ .

$- \frac{1}{4} \cdot \frac{1}{0 \cdot \sin (3 \cdot 0) + \cos^{3} (3 \cdot 0)}$

Step 9.4.6

Multiply $3$ by $0$ .

$- \frac{1}{4} \cdot \frac{1}{0 \cdot \sin (0) + \cos^{3} (3 \cdot 0)}$

Step 9.4.7

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

$- \frac{1}{4} \cdot \frac{1}{0 \cdot 0 + \cos^{3} (3 \cdot 0)}$

Step 9.4.8

Multiply $0$ by $0$ .

$- \frac{1}{4} \cdot \frac{1}{0 + \cos^{3} (3 \cdot 0)}$

Step 9.4.9

Multiply $3$ by $0$ .

$- \frac{1}{4} \cdot \frac{1}{0 + \cos^{3} (0)}$

Step 9.4.10

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

$- \frac{1}{4} \cdot \frac{1}{0 + 1^{3}}$

Step 9.4.11

One to any power is one.

$- \frac{1}{4} \cdot \frac{1}{0 + 1}$

Step 9.4.12

Add $0$ and $1$ .

$- \frac{1}{4} \cdot \frac{1}{1}$

Step 9.5

Cancel the common factor of $1$ .

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Step 9.5.1

Cancel the common factor.

$- \frac{1}{4} \cdot \frac{1}{1}$

Step 9.5.2

Rewrite the expression.

$- \frac{1}{4} \cdot 1$

Step 9.6

Multiply $- 1$ by $1$ .

$- \frac{1}{4}$