Calculus Examples

lim x \to 0 \frac{x}{arctan (2 x)}

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

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

\frac{lim x \to 0 x}{lim x \to 0 arctan (2 x)}

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

Step 1.1.2

Evaluate the limit of

x

$x$ by plugging in

0

$0$ for

x

$x$ .

\frac{0}{lim x \to 0 arctan (2 x)}

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit of

x

$x$ by plugging in

0

$0$ for

x

$x$ .

\frac{0}{0}

$\frac{0}{0}$

Step 1.1.3.2

Substitute

t

$t$ for

2 x

$2 x$ and let

t

$t$ approach

0

$0$ since

lim x \to 0 2 x = 0

$lim_{x \to 0} 2 x = 0$ .

\frac{0}{lim t \to 0 arctan (t)}

$\frac{0}{lim_{t \to 0} \arctan (t)}$

Step 1.1.3.3

Evaluate the limits by plugging in

0

$0$ for all occurrences of

t

$t$ .

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Step 1.1.3.3.1

Evaluate the limit of

x

$x$ by plugging in

0

$0$ for

x

$x$ .

\frac{0}{arctan (0)}

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

Step 1.1.3.3.2

The exact value of

arctan (0)

$\arctan (0)$ is

0

$0$ .

\frac{0}{0}

$\frac{0}{0}$

Step 1.1.3.3.3

The expression contains a division by

0

$0$ . The expression is undefined.

Undefined

\frac{0}{0}

$\frac{0}{0}$

Step 1.1.3.4

The expression contains a division by

0

$0$ . The expression is undefined.

Undefined

\frac{0}{0}

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by

0

$0$ . The expression is undefined.

Undefined

\frac{0}{0}

$\frac{0}{0}$

Step 1.2

Since

\frac{0}{0}

$\frac{0}{0}$ 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}{arctan (2 x)} = lim x \to 0 \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [arctan (2 x)]}

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

lim x \to 0 \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [arctan (2 x)]}

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

Step 1.3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

lim x \to 0 \frac{1}{\frac{d}{d x} [arctan (2 x)]}

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

Step 1.3.3

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = \arctan (x)$ and

$g (x) = 2 x$ .

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Step 1.3.3.1

To apply the Chain Rule, set

$u$ as

$2 x$ .

$lim_{x \to 0} \frac{1}{\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [2 x]}$

Step 1.3.3.2

The derivative of

$\arctan (u)$ with respect to

$u$ is

$\frac{1}{1 + u^{2}}$ .

$lim_{x \to 0} \frac{1}{\frac{1}{1 + u^{2}} \frac{d}{d x} [2 x]}$

Step 1.3.3.3

Replace all occurrences of

$u$ with

$2 x$ .

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

Step 1.3.4

Factor

$2$ out of

$2 x$ .

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

Step 1.3.5

Apply the product rule to

$2 (x)$ .

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

Step 1.3.6

Raise

$2$ to the power of

$2$ .

$lim_{x \to 0} \frac{1}{\frac{1}{1 + 4 x^{2}} \frac{d}{d x} [2 x]}$

Step 1.3.7

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 x$ with respect to

$x$ is

$2 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{1}{\frac{1}{1 + 4 x^{2}} (2 \frac{d}{d x} [x])}$

Step 1.3.8

Combine

$2$ and

$\frac{1}{1 + 4 x^{2}}$ .

$lim_{x \to 0} \frac{1}{\frac{2}{1 + 4 x^{2}} \frac{d}{d x} [x]}$

Step 1.3.9

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$lim_{x \to 0} \frac{1}{\frac{2}{1 + 4 x^{2}} \cdot 1}$

Step 1.3.10

Multiply

$\frac{2}{1 + 4 x^{2}}$ by

$1$ .

$lim_{x \to 0} \frac{1}{\frac{2}{1 + 4 x^{2}}}$

Step 1.3.11

Reorder terms.

$lim_{x \to 0} \frac{1}{\frac{2}{4 x^{2} + 1}}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} 1 \frac{4 x^{2} + 1}{2}$

Step 1.5

Multiply

$\frac{4 x^{2} + 1}{2}$ by

$1$ .

$lim_{x \to 0} \frac{4 x^{2} + 1}{2}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term

$\frac{1}{2}$ outside of the limit because it is constant with respect to

$x$ .

$\frac{1}{2} lim_{x \to 0} 4 x^{2} + 1$

Step 2.2

Split the limit using the Sum of Limits Rule on the limit as

$x$ approaches

$0$ .

$\frac{1}{2} (lim_{x \to 0} 4 x^{2} + lim_{x \to 0} 1)$

Step 2.3

Move the term

$4$ outside of the limit because it is constant with respect to

$x$ .

$\frac{1}{2} (4 lim_{x \to 0} x^{2} + lim_{x \to 0} 1)$

Step 2.4

Move the exponent

$2$ from

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

$\frac{1}{2} (4 {(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 1)$

Step 2.5

Evaluate the limit of

$1$ which is constant as

$x$ approaches

$0$ .

$\frac{1}{2} (4 {(lim_{x \to 0} x)}^{2} + 1)$

Step 3

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

$\frac{1}{2} (4 \cdot 0^{2} + 1)$

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raising

$0$ to any positive power yields

$0$ .

$\frac{1}{2} (4 \cdot 0 + 1)$

Step 4.1.2

Multiply

$4$ by

$0$ .

$\frac{1}{2} (0 + 1)$

Step 4.2

Add

$0$ and

$1$ .

$\frac{1}{2} \cdot 1$

Step 4.3

Multiply

$\frac{1}{2}$ by

$1$ .

$\frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$