Calculus Examples

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

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.

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

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

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

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

Step 1.2.1.2

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

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

Step 1.2.2

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

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

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

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 limit inside the trig function because sine is continuous.

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

Step 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 1.3.2

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

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

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Multiply $2$ by $0$ .

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

Step 1.3.3.2

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

$\frac{0}{0}$

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 3.3

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

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

Step 3.4

Simplify.

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Step 3.4.1

Reorder the factors of $\sec^{2} (x^{2}) (2 x)$ .

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

Step 3.4.2

Rewrite $\sec (x^{2})$ in terms of sines and cosines.

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

Step 3.4.3

Apply the product rule to $\frac{1}{\cos (x^{2})}$ .

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

Step 3.4.4

One to any power is one.

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

Step 3.4.5

Multiply $2 x \frac{1}{\cos^{2} (x^{2})}$ .

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Step 3.4.5.1

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

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

Step 3.4.5.2

Combine $x$ and $\frac{2}{\cos^{2} (x^{2})}$ .

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

Step 3.4.6

Move $2$ to the left of $x$ .

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

Step 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 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Multiply $2$ by $1$ .

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

Step 3.9

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

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

Step 3.10

Multiply $2$ by $\cos (2 x)$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Simplify terms.

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Step 5.1

Multiply $\frac{2 x}{\cos^{2} (x^{2})}$ by $\frac{1}{2 \cos (2 x)}$ .

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

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

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

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

Step 14

Simplify the answer.

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Step 14.1

Multiply by $1$ .

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

Step 14.2

Separate fractions.

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

Step 14.3

Convert from $\frac{1}{\cos^{2} (0^{2})}$ to $\sec^{2} (0^{2})$ .

$\frac{0}{1 \cdot \cos (2 \cdot 0)} \sec^{2} (0^{2})$

Step 14.4

Multiply $2$ by $0$ .

$\frac{0}{1 \cdot \cos (0)} \sec^{2} (0^{2})$

Step 14.5

Multiply $\cos (0)$ by $1$ .

$\frac{0}{\cos (0)} \sec^{2} (0^{2})$

Step 14.6

Separate fractions.

$\frac{0}{1} \cdot \frac{1}{\cos (0)} \sec^{2} (0^{2})$

Step 14.7

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

$\frac{0}{1} \sec (0) \sec^{2} (0^{2})$

Step 14.8

Divide $0$ by $1$ .

$0 \sec (0) \sec^{2} (0^{2})$

Step 14.9

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

$0 \cdot 1 \sec^{2} (0^{2})$

Step 14.10

Multiply $0$ by $1$ .

$0 \sec^{2} (0^{2})$

Step 14.11

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

$0 \sec^{2} (0)$

Step 14.12

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

$0 \cdot 1^{2}$

Step 14.13

One to any power is one.

$0 \cdot 1$

Step 14.14

Multiply $0$ by $1$ .

$0$