Calculus Examples

$lim_{x \to 0} \frac{x}{\arctan (8 x)}$

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.

$\frac{lim_{x \to 0} x}{lim_{x \to 0} \arctan (8 x)}$

Step 1.1.2

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

$\frac{0}{lim_{x \to 0} \arctan (8 x)}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

$\frac{0}{0}$

Step 1.1.3.2

Substitute $t$ for $8 x$ and let $t$ approach $0$ since $lim_{x \to 0} 8 x = 0$ .

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

Step 1.1.3.3

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

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Step 1.1.3.3.1

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

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

Step 1.1.3.3.2

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

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 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.

$lim_{x \to 0} \frac{x}{\arctan (8 x)} = lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [\arctan (8 x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [\arctan (8 x)]}$

Step 1.3.2

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

$lim_{x \to 0} \frac{1}{\frac{d}{d x} [\arctan (8 x)]}$

Step 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 $f (x) = \arctan (x)$ and $g (x) = 8 x$ .

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Step 1.3.3.1

To apply the Chain Rule, set $u$ as $8 x$ .

$lim_{x \to 0} \frac{1}{\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [8 x]}$

Step 1.3.3.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

$lim_{x \to 0} \frac{1}{\frac{1}{1 + u^{2}} \frac{d}{d x} [8 x]}$

Step 1.3.3.3

Replace all occurrences of $u$ with $8 x$ .

$lim_{x \to 0} \frac{1}{\frac{1}{1 + {(8 x)}^{2}} \frac{d}{d x} [8 x]}$

Step 1.3.4

Factor $8$ out of $8 x$ .

$lim_{x \to 0} \frac{1}{\frac{1}{1 + {(8 (x))}^{2}} \frac{d}{d x} [8 x]}$

Step 1.3.5

Apply the product rule to $8 (x)$ .

$lim_{x \to 0} \frac{1}{\frac{1}{1 + 8^{2} x^{2}} \frac{d}{d x} [8 x]}$

Step 1.3.6

Raise $8$ to the power of $2$ .

$lim_{x \to 0} \frac{1}{\frac{1}{1 + 64 x^{2}} \frac{d}{d x} [8 x]}$

Step 1.3.7

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{1}{\frac{1}{1 + 64 x^{2}} (8 \frac{d}{d x} [x])}$

Step 1.3.8

Combine $8$ and $\frac{1}{1 + 64 x^{2}}$ .

$lim_{x \to 0} \frac{1}{\frac{8}{1 + 64 x^{2}} \frac{d}{d x} [x]}$

Step 1.3.9

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

$lim_{x \to 0} \frac{1}{\frac{8}{1 + 64 x^{2}} \cdot 1}$

Step 1.3.10

Multiply $\frac{8}{1 + 64 x^{2}}$ by $1$ .

$lim_{x \to 0} \frac{1}{\frac{8}{1 + 64 x^{2}}}$

Step 1.3.11

Reorder terms.

$lim_{x \to 0} \frac{1}{\frac{8}{64 x^{2} + 1}}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} 1 \frac{64 x^{2} + 1}{8}$

Step 1.5

Multiply $\frac{64 x^{2} + 1}{8}$ by $1$ .

$lim_{x \to 0} \frac{64 x^{2} + 1}{8}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{8}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{8} lim_{x \to 0} 64 x^{2} + 1$

Step 2.2

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

$\frac{1}{8} (lim_{x \to 0} 64 x^{2} + lim_{x \to 0} 1)$

Step 2.3

Move the term $64$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{8} (64 lim_{x \to 0} x^{2} + lim_{x \to 0} 1)$

Step 2.4

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

$\frac{1}{8} (64 {(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 1)$

Step 2.5

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{1}{8} (64 {(lim_{x \to 0} x)}^{2} + 1)$

Step 3

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

$\frac{1}{8} (64 \cdot 0^{2} + 1)$

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 $0$ .

$\frac{1}{8} (64 \cdot 0 + 1)$

Step 4.1.2

Multiply $64$ by $0$ .

$\frac{1}{8} (0 + 1)$

Step 4.2

Add $0$ and $1$ .

$\frac{1}{8} \cdot 1$

Step 4.3

Multiply $\frac{1}{8}$ by $1$ .

$\frac{1}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{8}$

Decimal Form:

$0.125$