Calculus Examples

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

Step 1

Convert from $\frac{1}{\sin^{2} (3 x)}$ to $\csc^{2} (3 x)$ .

$lim_{x \to 0} x^{2} \csc^{2} (3 x)$

Step 2

Rewrite $x^{2} \csc^{2} (3 x)$ as $\frac{x^{2}}{\csc^{- 2} (3 x)}$ .

$lim_{x \to 0} \frac{x^{2}}{\csc^{- 2} (3 x)}$

Step 3

Set up the limit as a left-sided limit.

$lim_{x \to 0^{-}} \frac{x^{2}}{\csc^{- 2} (3 x)}$

Step 4

Evaluate the left-sided limit.

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Step 4.1

Apply L'Hospital's rule.

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Step 4.1.1

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

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Step 4.1.1.1

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

$\frac{lim_{x \to 0^{-}} x^{2}}{lim_{x \to 0^{-}} \csc^{- 2} (3 x)}$

Step 4.1.1.2

Evaluate the limit of the numerator.

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Step 4.1.1.2.1

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

$\frac{{(lim_{x \to 0^{-}} x)}^{2}}{lim_{x \to 0^{-}} \csc^{- 2} (3 x)}$

Step 4.1.1.2.2

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

$\frac{0^{2}}{lim_{x \to 0^{-}} \csc^{- 2} (3 x)}$

Step 4.1.1.2.3

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

$\frac{0}{lim_{x \to 0^{-}} \csc^{- 2} (3 x)}$

Step 4.1.1.3

Evaluate the limit of the denominator.

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Step 4.1.1.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.1.1.3.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{\csc^{2} (3 x)}$ approaches $0$ .

$\frac{0}{0}$

Step 4.1.1.3.3

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

Undefined

$\frac{0}{0}$

Step 4.1.1.4

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

Undefined

$\frac{0}{0}$

Step 4.1.2

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.

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

Step 4.1.3

Find the derivative of the numerator and denominator.

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Step 4.1.3.1

Differentiate the numerator and denominator.

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

Step 4.1.3.2

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

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

Step 4.1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 2}$ and $g (x) = \csc (3 x)$ .

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Step 4.1.3.3.1

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

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

Step 4.1.3.3.2

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

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

Step 4.1.3.3.3

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

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

Step 4.1.3.4

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

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Step 4.1.3.4.1

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

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

Step 4.1.3.4.2

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

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

Step 4.1.3.4.3

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

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

Step 4.1.3.5

Multiply $- 1$ by $- 2$ .

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

Step 4.1.3.6

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

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

Step 4.1.3.7

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

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

Step 4.1.3.8

Subtract $3$ from $1$ .

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

Step 4.1.3.9

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

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

Step 4.1.3.10

Multiply $3$ by $2$ .

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

Step 4.1.3.11

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

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

Step 4.1.3.12

Multiply $6$ by $1$ .

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

Step 4.1.3.13

Simplify.

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Step 4.1.3.13.1

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

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

Step 4.1.3.13.2

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 4.1.3.13.3

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

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

Step 4.1.3.13.4

Cancel the common factor of $\sin (3 x)$ .

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Step 4.1.3.13.4.1

Factor $\sin (3 x)$ out of $6 \sin^{2} (3 x)$ .

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

Step 4.1.3.13.4.2

Cancel the common factor.

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

Step 4.1.3.13.4.3

Rewrite the expression.

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

Step 4.1.4

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

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Step 4.1.4.1

Factor $2$ out of $2 x$ .

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

Step 4.1.4.2

Cancel the common factors.

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Step 4.1.4.2.1

Factor $2$ out of $6 \sin (3 x) \cos (3 x)$ .

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

Step 4.1.4.2.2

Cancel the common factor.

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

Step 4.1.4.2.3

Rewrite the expression.

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

Step 4.1.5

Separate fractions.

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

Step 4.1.6

Convert from $\frac{1}{\sin (3 x)}$ to $\csc (3 x)$ .

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

Step 4.1.7

Separate fractions.

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

Step 4.1.8

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

$lim_{x \to 0^{-}} \frac{x}{3} \sec (3 x) \csc (3 x)$

Step 4.1.9

Combine $\frac{x}{3}$ and $\sec (3 x)$ .

$lim_{x \to 0^{-}} \frac{x \sec (3 x)}{3} \csc (3 x)$

Step 4.1.10

Combine $\frac{x \sec (3 x)}{3}$ and $\csc (3 x)$ .

$lim_{x \to 0^{-}} \frac{x \sec (3 x) \csc (3 x)}{3}$

Step 4.2

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

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

Step 4.3

Make a table to show the behavior of the function $x \sec (3 x) \csc (3 x)$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & x \sec (3 x) \csc (3 x) \\ - 0.1 & 0.35420643 \\ - 0.01 & 0.33353341 \\ - 0.001 & 0.33333533 \\ - 0.0001 & 0.33333335 \end{array}$

Step 4.4

As the $x$ values approach $0$ , the function values approach $0.333$ . Thus, the limit of $x \sec (3 x) \csc (3 x)$ as $x$ approaches $0$ from the left is $0.333$ .

$\frac{1}{3} \cdot 0.333$

Step 4.5

Simplify the answer.

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Step 4.5.1

Combine $\frac{1}{3}$ and $0.333$ .

$\frac{0.333}{3}$

Step 4.5.2

Divide $0.333$ by $3$ .

$0.111$

Step 5

Set up the limit as a right-sided limit.

$lim_{x \to 0^{+}} \frac{x^{2}}{\csc^{- 2} (3 x)}$

Step 6

Evaluate the right-sided limit.

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Step 6.1

Apply L'Hospital's rule.

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Step 6.1.1

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

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Step 6.1.1.1

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

$\frac{lim_{x \to 0^{+}} x^{2}}{lim_{x \to 0^{+}} \csc^{- 2} (3 x)}$

Step 6.1.1.2

Evaluate the limit of the numerator.

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Step 6.1.1.2.1

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

$\frac{{(lim_{x \to 0^{+}} x)}^{2}}{lim_{x \to 0^{+}} \csc^{- 2} (3 x)}$

Step 6.1.1.2.2

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

$\frac{0^{2}}{lim_{x \to 0^{+}} \csc^{- 2} (3 x)}$

Step 6.1.1.2.3

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

$\frac{0}{lim_{x \to 0^{+}} \csc^{- 2} (3 x)}$

Step 6.1.1.3

Evaluate the limit of the denominator.

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Step 6.1.1.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{0}{lim_{x \to 0^{+}} \frac{1}{\csc^{2} (3 x)}}$

Step 6.1.1.3.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{\csc^{2} (3 x)}$ approaches $0$ .

$\frac{0}{0}$

Step 6.1.1.3.3

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

Undefined

$\frac{0}{0}$

Step 6.1.1.4

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

Undefined

$\frac{0}{0}$

Step 6.1.2

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.

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

Step 6.1.3

Find the derivative of the numerator and denominator.

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Step 6.1.3.1

Differentiate the numerator and denominator.

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

Step 6.1.3.2

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

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

Step 6.1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 2}$ and $g (x) = \csc (3 x)$ .

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Step 6.1.3.3.1

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

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

Step 6.1.3.3.2

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

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

Step 6.1.3.3.3

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

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

Step 6.1.3.4

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

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Step 6.1.3.4.1

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

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

Step 6.1.3.4.2

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

$lim_{x \to 0^{+}} \frac{2 x}{- 2 \csc^{- 3} (3 x) (- \csc (u_{2}) \cot (u_{2}) \frac{d}{d x} [3 x])}$

Step 6.1.3.4.3

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

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

Step 6.1.3.5

Multiply $- 1$ by $- 2$ .

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

Step 6.1.3.6

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

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

Step 6.1.3.7

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

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

Step 6.1.3.8

Subtract $3$ from $1$ .

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

Step 6.1.3.9

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

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

Step 6.1.3.10

Multiply $3$ by $2$ .

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

Step 6.1.3.11

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

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

Step 6.1.3.12

Multiply $6$ by $1$ .

$lim_{x \to 0^{+}} \frac{2 x}{6 \csc^{- 2} (3 x) \cot (3 x)}$

Step 6.1.3.13

Simplify.

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Step 6.1.3.13.1

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

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

Step 6.1.3.13.2

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 6.1.3.13.3

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

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

Step 6.1.3.13.4

Cancel the common factor of $\sin (3 x)$ .

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Step 6.1.3.13.4.1

Factor $\sin (3 x)$ out of $6 \sin^{2} (3 x)$ .

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

Step 6.1.3.13.4.2

Cancel the common factor.

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

Step 6.1.3.13.4.3

Rewrite the expression.

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

Step 6.1.4

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

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Step 6.1.4.1

Factor $2$ out of $2 x$ .

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

Step 6.1.4.2

Cancel the common factors.

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Step 6.1.4.2.1

Factor $2$ out of $6 \sin (3 x) \cos (3 x)$ .

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

Step 6.1.4.2.2

Cancel the common factor.

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

Step 6.1.4.2.3

Rewrite the expression.

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

Step 6.1.5

Separate fractions.

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

Step 6.1.6

Convert from $\frac{1}{\sin (3 x)}$ to $\csc (3 x)$ .

$lim_{x \to 0^{+}} \frac{x}{3 \cos (3 x)} \csc (3 x)$

Step 6.1.7

Separate fractions.

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

Step 6.1.8

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

$lim_{x \to 0^{+}} \frac{x}{3} \sec (3 x) \csc (3 x)$

Step 6.1.9

Combine $\frac{x}{3}$ and $\sec (3 x)$ .

$lim_{x \to 0^{+}} \frac{x \sec (3 x)}{3} \csc (3 x)$

Step 6.1.10

Combine $\frac{x \sec (3 x)}{3}$ and $\csc (3 x)$ .

$lim_{x \to 0^{+}} \frac{x \sec (3 x) \csc (3 x)}{3}$

Step 6.2

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

$\frac{1}{3} lim_{x \to 0^{+}} x \sec (3 x) \csc (3 x)$

Step 6.3

Make a table to show the behavior of the function $x \sec (3 x) \csc (3 x)$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & x \sec (3 x) \csc (3 x) \\ 0.1 & 0.35420643 \\ 0.01 & 0.33353341 \\ 0.001 & 0.33333533 \\ 0.0001 & 0.33333335 \end{array}$

Step 6.4

As the $x$ values approach $0$ , the function values approach $0.333$ . Thus, the limit of $x \sec (3 x) \csc (3 x)$ as $x$ approaches $0$ from the right is $0.333$ .

$\frac{1}{3} \cdot 0.333$

Step 6.5

Simplify the answer.

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Step 6.5.1

Combine $\frac{1}{3}$ and $0.333$ .

$\frac{0.333}{3}$

Step 6.5.2

Divide $0.333$ by $3$ .

$0.111$

Step 7

Since the left-sided limit is equal to the right-sided limit, the limit is equal to $0.111$ .

$0.111$