Calculus Examples

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

Step 1

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

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

Step 2

Set up the limit as a left-sided limit.

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

Step 3

Evaluate the left-sided limit.

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Step 3.1

Apply L'Hospital's rule.

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Step 3.1.1

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

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Step 3.1.1.1

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

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

Step 3.1.1.2

Evaluate the limit of the numerator.

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Step 3.1.1.2.1

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

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

Step 3.1.1.2.2

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

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

Step 3.1.1.2.3

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

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

Step 3.1.1.3

Evaluate the limit of the denominator.

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Step 3.1.1.3.1

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

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

Step 3.1.1.3.2

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

$\frac{0}{0}$

Step 3.1.1.3.3

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

Undefined

$\frac{0}{0}$

Step 3.1.1.4

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

Undefined

$\frac{0}{0}$

Step 3.1.2

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.

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

Step 3.1.3

Find the derivative of the numerator and denominator.

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Step 3.1.3.1

Differentiate the numerator and denominator.

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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 $g (x) = \csc (x)$ .

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Step 3.1.3.3.1

To apply the Chain Rule, set $u$ as $\csc (x)$ .

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

Step 3.1.3.3.2

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

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

Step 3.1.3.3.3

Replace all occurrences of $u$ with $\csc (x)$ .

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

Step 3.1.3.4

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

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

Step 3.1.3.5

Multiply $- 1$ by $- 2$ .

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

Step 3.1.3.6

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

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

Step 3.1.3.7

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

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

Step 3.1.3.8

Subtract $3$ from $1$ .

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

Step 3.1.3.9

Simplify.

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Step 3.1.3.9.1

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

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

Step 3.1.3.9.2

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

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

Step 3.1.3.9.3

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

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

Step 3.1.3.9.4

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

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Step 3.1.3.9.4.1

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

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

Step 3.1.3.9.4.2

Cancel the common factor.

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

Step 3.1.3.9.4.3

Rewrite the expression.

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

Step 3.1.3.9.5

Apply the sine double-angle identity.

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

Step 3.2

The limit of $\frac{2 x}{\sin (2 x)}$ as $x$ approaches $0$ is $1$ .

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

Evaluate the limit of the numerator.

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Step 3.2.1.2.1

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 3.2.1.2.2

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

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

Step 3.2.1.2.3

Multiply $2$ by $0$ .

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

Step 3.2.1.3

Evaluate the limit of the denominator.

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Step 3.2.1.3.1

Evaluate the limit.

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Step 3.2.1.3.1.1

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

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

Step 3.2.1.3.1.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 3.2.1.3.2

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

$\frac{0}{\sin (2 \cdot 0)}$

Step 3.2.1.3.3

Simplify the answer.

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Step 3.2.1.3.3.1

Multiply $2$ by $0$ .

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

Step 3.2.1.3.3.2

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

$\frac{0}{0}$

Step 3.2.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 3.2.1.3.4

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

Undefined

$\frac{0}{0}$

Step 3.2.1.4

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

Undefined

$\frac{0}{0}$

Step 3.2.2

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.

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

Step 3.2.3

Find the derivative of the numerator and denominator.

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Step 3.2.3.1

Differentiate the numerator and denominator.

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

Step 3.2.3.2

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

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

Step 3.2.3.3

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

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

Step 3.2.3.4

Multiply $2$ by $1$ .

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

Step 3.2.3.5

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 $g (x) = 2 x$ .

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Step 3.2.3.5.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 3.2.3.5.2

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

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

Step 3.2.3.5.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 3.2.3.6

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

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

Step 3.2.3.7

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

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

Step 3.2.3.8

Multiply $2$ by $1$ .

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

Step 3.2.3.9

Move $2$ to the left of $\cos (2 x)$ .

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

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

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

Step 3.2.4.2

Rewrite the expression.

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

Step 3.2.5

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

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

Step 3.2.6

Evaluate the limit.

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Step 3.2.6.1

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

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

Step 3.2.6.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 3.2.7

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

$\sec (2 \cdot 0)$

Step 3.2.8

Simplify the answer.

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Step 3.2.8.1

Multiply $2$ by $0$ .

$\sec (0)$

Step 3.2.8.2

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

$1$

Step 4

Set up the limit as a right-sided limit.

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

Step 5

Evaluate the right-sided limit.

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Step 5.1

Apply L'Hospital's rule.

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Step 5.1.1

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

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Step 5.1.1.1

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

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

Step 5.1.1.2

Evaluate the limit of the numerator.

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Step 5.1.1.2.1

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

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

Step 5.1.1.2.2

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

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

Step 5.1.1.2.3

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

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

Step 5.1.1.3

Evaluate the limit of the denominator.

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Step 5.1.1.3.1

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

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

Step 5.1.1.3.2

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

$\frac{0}{0}$

Step 5.1.1.3.3

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

Undefined

$\frac{0}{0}$

Step 5.1.1.4

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

Undefined

$\frac{0}{0}$

Step 5.1.2

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.

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

Step 5.1.3

Find the derivative of the numerator and denominator.

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Step 5.1.3.1

Differentiate the numerator and denominator.

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

Step 5.1.3.2

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

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

Step 5.1.3.3

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 $g (x) = \csc (x)$ .

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Step 5.1.3.3.1

To apply the Chain Rule, set $u$ as $\csc (x)$ .

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

Step 5.1.3.3.2

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

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

Step 5.1.3.3.3

Replace all occurrences of $u$ with $\csc (x)$ .

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

Step 5.1.3.4

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

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

Step 5.1.3.5

Multiply $- 1$ by $- 2$ .

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

Step 5.1.3.6

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

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

Step 5.1.3.7

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

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

Step 5.1.3.8

Subtract $3$ from $1$ .

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

Step 5.1.3.9

Simplify.

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Step 5.1.3.9.1

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

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

Step 5.1.3.9.2

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

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

Step 5.1.3.9.3

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

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

Step 5.1.3.9.4

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

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Step 5.1.3.9.4.1

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

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

Step 5.1.3.9.4.2

Cancel the common factor.

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

Step 5.1.3.9.4.3

Rewrite the expression.

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

Step 5.1.3.9.5

Apply the sine double-angle identity.

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

Step 5.2

The limit of $\frac{2 x}{\sin (2 x)}$ as $x$ approaches $0$ is $1$ .

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Step 5.2.1

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

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Step 5.2.1.1

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

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

Step 5.2.1.2

Evaluate the limit of the numerator.

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Step 5.2.1.2.1

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 5.2.1.2.2

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

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

Step 5.2.1.2.3

Multiply $2$ by $0$ .

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

Step 5.2.1.3

Evaluate the limit of the denominator.

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Step 5.2.1.3.1

Evaluate the limit.

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Step 5.2.1.3.1.1

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

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

Step 5.2.1.3.1.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 5.2.1.3.2

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

$\frac{0}{\sin (2 \cdot 0)}$

Step 5.2.1.3.3

Simplify the answer.

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Step 5.2.1.3.3.1

Multiply $2$ by $0$ .

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

Step 5.2.1.3.3.2

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

$\frac{0}{0}$

Step 5.2.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 5.2.1.3.4

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

Undefined

$\frac{0}{0}$

Step 5.2.1.4

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

Undefined

$\frac{0}{0}$

Step 5.2.2

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.

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

Step 5.2.3

Find the derivative of the numerator and denominator.

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Step 5.2.3.1

Differentiate the numerator and denominator.

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

Step 5.2.3.2

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

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

Step 5.2.3.3

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

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

Step 5.2.3.4

Multiply $2$ by $1$ .

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

Step 5.2.3.5

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 $g (x) = 2 x$ .

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Step 5.2.3.5.1

To apply the Chain Rule, set $u$ as $2 x$ .

$lim_{x \to 0^{+}} \frac{2}{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [2 x]}$

Step 5.2.3.5.2

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

$lim_{x \to 0^{+}} \frac{2}{\cos (u) \frac{d}{d x} [2 x]}$

Step 5.2.3.5.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 5.2.3.6

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

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

Step 5.2.3.7

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

$lim_{x \to 0^{+}} \frac{2}{\cos (2 x) (2 \cdot 1)}$

Step 5.2.3.8

Multiply $2$ by $1$ .

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

Step 5.2.3.9

Move $2$ to the left of $\cos (2 x)$ .

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

Step 5.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.1

Cancel the common factor.

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

Step 5.2.4.2

Rewrite the expression.

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

Step 5.2.5

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

$lim_{x \to 0^{+}} \sec (2 x)$

Step 5.2.6

Evaluate the limit.

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Step 5.2.6.1

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

$\sec (lim_{x \to 0^{+}} 2 x)$

Step 5.2.6.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\sec (2 lim_{x \to 0^{+}} x)$

Step 5.2.7

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

$\sec (2 \cdot 0)$

Step 5.2.8

Simplify the answer.

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Step 5.2.8.1

Multiply $2$ by $0$ .

$\sec (0)$

Step 5.2.8.2

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

$1$

Step 6

Since the left-sided limit is equal to the right-sided limit, the limit is equal to $1$ .

$1$