Calculus Examples

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

Step 1

Move the term

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

$x$ .

$\frac{1}{3} lim_{x \to 0} \frac{\arctan (2 x)}{x}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{1}{3} \cdot \frac{lim_{x \to 0} \arctan (2 x)}{lim_{x \to 0} x}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

$\frac{1}{3} \cdot \frac{0}{lim_{x \to 0} x}$

Step 2.1.2.2

Substitute

$t$ for

$2 x$ and let

$t$ approach

$0$ since

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

$\frac{1}{3} \cdot \frac{lim_{t \to 0} \arctan (t)}{lim_{x \to 0} x}$

Step 2.1.2.3

Evaluate the limits by plugging in

$0$ for all occurrences of

$t$ .

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Step 2.1.2.3.1

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

$\frac{1}{3} \cdot \frac{\arctan (0)}{lim_{x \to 0} x}$

Step 2.1.2.3.2

The exact value of

$\arctan (0)$ is

$0$ .

$\frac{1}{3} \cdot \frac{0}{lim_{x \to 0} x}$

Step 2.1.3

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

$\frac{1}{3} \cdot \frac{0}{0}$

Step 2.1.4

The expression contains a division by

$0$ . The expression is undefined.

Undefined

$\frac{1}{3} \cdot \frac{0}{0}$

Step 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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$\frac{1}{3} lim_{x \to 0} \frac{\frac{d}{d x} [\arctan (2 x)]}{\frac{d}{d x} [x]}$

Step 2.3.2

Differentiate using the chain rule, which states that

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

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

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

$g (x) = 2 x$ .

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Step 2.3.2.1

To apply the Chain Rule, set

$u$ as

$2 x$ .

$\frac{1}{3} lim_{x \to 0} \frac{\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [2 x]}{\frac{d}{d x} [x]}$

Step 2.3.2.2

The derivative of

$\arctan (u)$ with respect to

$u$ is

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

$\frac{1}{3} lim_{x \to 0} \frac{\frac{1}{1 + u^{2}} \frac{d}{d x} [2 x]}{\frac{d}{d x} [x]}$

Step 2.3.2.3

Replace all occurrences of

$u$ with

$2 x$ .

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

Step 2.3.3

Factor

$2$ out of

$2 x$ .

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

Step 2.3.4

Apply the product rule to

$2 (x)$ .

$\frac{1}{3} lim_{x \to 0} \frac{\frac{1}{1 + 2^{2} x^{2}} \frac{d}{d x} [2 x]}{\frac{d}{d x} [x]}$

Step 2.3.5

Raise

$2$ to the power of

$2$ .

$\frac{1}{3} lim_{x \to 0} \frac{\frac{1}{1 + 4 x^{2}} \frac{d}{d x} [2 x]}{\frac{d}{d x} [x]}$

Step 2.3.6

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 x$ with respect to

$x$ is

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

$\frac{1}{3} lim_{x \to 0} \frac{\frac{1}{1 + 4 x^{2}} \cdot 2 \frac{d}{d x} [x]}{\frac{d}{d x} [x]}$

Step 2.3.7

Combine

$2$ and

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

$\frac{1}{3} lim_{x \to 0} \frac{\frac{2}{1 + 4 x^{2}} \frac{d}{d x} [x]}{\frac{d}{d x} [x]}$

Step 2.3.8

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

$\frac{1}{3} lim_{x \to 0} \frac{\frac{2}{1 + 4 x^{2}} \cdot 1}{\frac{d}{d x} [x]}$

Step 2.3.9

Multiply

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

$1$ .

$\frac{1}{3} lim_{x \to 0} \frac{\frac{2}{1 + 4 x^{2}}}{\frac{d}{d x} [x]}$

Step 2.3.10

Reorder terms.

$\frac{1}{3} lim_{x \to 0} \frac{\frac{2}{4 x^{2} + 1}}{\frac{d}{d x} [x]}$

Step 2.3.11

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

$\frac{1}{3} lim_{x \to 0} \frac{\frac{2}{4 x^{2} + 1}}{1}$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{3} lim_{x \to 0} \frac{2}{4 x^{2} + 1} \cdot 1$

Step 2.5

Multiply

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

$1$ .

$\frac{1}{3} lim_{x \to 0} \frac{2}{4 x^{2} + 1}$

Step 3

Evaluate the limit.

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Step 3.1

Move the term

$2$ outside of the limit because it is constant with respect to

$x$ .

$\frac{1}{3} \cdot 2 lim_{x \to 0} \frac{1}{4 x^{2} + 1}$

Step 3.2

Split the limit using the Limits Quotient Rule on the limit as

$x$ approaches

$0$ .

$\frac{1}{3} \cdot 2 \frac{lim_{x \to 0} 1}{lim_{x \to 0} 4 x^{2} + 1}$

Step 3.3

Evaluate the limit of

$1$ which is constant as

$x$ approaches

$0$ .

$\frac{1}{3} \cdot 2 \frac{1}{lim_{x \to 0} 4 x^{2} + 1}$

Step 3.4

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

$x$ approaches

$0$ .

$\frac{1}{3} \cdot 2 \frac{1}{lim_{x \to 0} 4 x^{2} + lim_{x \to 0} 1}$

Step 3.5

Move the term

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

$x$ .

$\frac{1}{3} \cdot 2 \frac{1}{4 lim_{x \to 0} x^{2} + lim_{x \to 0} 1}$

Step 3.6

Move the exponent

$2$ from

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

$\frac{1}{3} \cdot 2 \frac{1}{4 {(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 1}$

Step 3.7

Evaluate the limit of

$1$ which is constant as

$x$ approaches

$0$ .

$\frac{1}{3} \cdot 2 \frac{1}{4 {(lim_{x \to 0} x)}^{2} + 1}$

Step 4

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

$\frac{1}{3} \cdot 2 \frac{1}{4 \cdot 0^{2} + 1}$

Step 5

Simplify the answer.

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Step 5.1

Combine

$\frac{1}{3}$ and

$2$ .

$\frac{2}{3} \cdot \frac{1}{4 \cdot 0^{2} + 1}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Raising

$0$ to any positive power yields

$0$ .

$\frac{2}{3} \cdot \frac{1}{4 \cdot 0 + 1}$

Step 5.2.2

Multiply

$4$ by

$0$ .

$\frac{2}{3} \cdot \frac{1}{0 + 1}$

Step 5.2.3

Add

$0$ and

$1$ .

$\frac{2}{3} \cdot \frac{1}{1}$

Step 5.3

Cancel the common factor of

$1$ .

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Step 5.3.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{1}{1}$

Step 5.3.2

Rewrite the expression.

$\frac{2}{3} \cdot 1$

Step 5.4

Multiply

$\frac{2}{3}$ by

$1$ .

$\frac{2}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$