Calculus Examples

$lim_{x \to 0} \frac{\arcsin (9 x)}{x}$

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.

$\frac{lim_{x \to 0} \arcsin (9 x)}{lim_{x \to 0} x}$

Step 1.2

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

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Step 1.2.1

Evaluate the limit of $\arcsin (9 x)$ by plugging in $0$ for $x$ .

$\frac{\arcsin (9 \cdot 0)}{lim_{x \to 0} x}$

Step 1.2.2

Multiply $9$ by $0$ .

$\frac{\arcsin (0)}{lim_{x \to 0} x}$

Step 1.2.3

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

$\frac{0}{lim_{x \to 0} x}$

Step 1.3

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

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{\arcsin (9 x)}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\arcsin (9 x)]}{\frac{d}{d x} [x]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\arcsin (9 x)]}{\frac{d}{d x} [x]}$

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 $f (x) = \arcsin (x)$ and $g (x) = 9 x$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $9 x$ .

$lim_{x \to 0} \frac{\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [9 x]}{\frac{d}{d x} [x]}$

Step 3.2.2

The derivative of $\arcsin (u)$ with respect to $u$ is $\frac{1}{\sqrt{1 - u^{2}}}$ .

$lim_{x \to 0} \frac{\frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [9 x]}{\frac{d}{d x} [x]}$

Step 3.2.3

Replace all occurrences of $u$ with $9 x$ .

$lim_{x \to 0} \frac{\frac{1}{\sqrt{1 - {(9 x)}^{2}}} \frac{d}{d x} [9 x]}{\frac{d}{d x} [x]}$

Step 3.3

Factor $9$ out of $9 x$ .

$lim_{x \to 0} \frac{\frac{1}{\sqrt{1 - {(9 (x))}^{2}}} \frac{d}{d x} [9 x]}{\frac{d}{d x} [x]}$

Step 3.4

Apply the product rule to $9 (x)$ .

$lim_{x \to 0} \frac{\frac{1}{\sqrt{1 - (9^{2} x^{2})}} \frac{d}{d x} [9 x]}{\frac{d}{d x} [x]}$

Step 3.5

Raise $9$ to the power of $2$ .

$lim_{x \to 0} \frac{\frac{1}{\sqrt{1 - (81 x^{2})}} \frac{d}{d x} [9 x]}{\frac{d}{d x} [x]}$

Step 3.6

Multiply $81$ by $- 1$ .

$lim_{x \to 0} \frac{\frac{1}{\sqrt{1 - 81 x^{2}}} \frac{d}{d x} [9 x]}{\frac{d}{d x} [x]}$

Step 3.7

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\frac{1}{\sqrt{1 - 81 x^{2}}} (9 \frac{d}{d x} [x])}{\frac{d}{d x} [x]}$

Step 3.8

Combine $9$ and $\frac{1}{\sqrt{1 - 81 x^{2}}}$ .

$lim_{x \to 0} \frac{\frac{9}{\sqrt{1 - 81 x^{2}}} \frac{d}{d x} [x]}{\frac{d}{d x} [x]}$

Step 3.9

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

$lim_{x \to 0} \frac{\frac{9}{\sqrt{1 - 81 x^{2}}} \cdot 1}{\frac{d}{d x} [x]}$

Step 3.10

Multiply $\frac{9}{\sqrt{1 - 81 x^{2}}}$ by $1$ .

$lim_{x \to 0} \frac{\frac{9}{\sqrt{1 - 81 x^{2}}}}{\frac{d}{d x} [x]}$

Step 3.11

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

$lim_{x \to 0} \frac{\frac{9}{\sqrt{1 - 81 x^{2}}}}{1}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{9}{\sqrt{1 - 81 x^{2}}} \cdot 1$

Step 5

Evaluate the limit.

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Step 5.1

Multiply $\frac{9}{\sqrt{1 - 81 x^{2}}}$ by $1$ .

$lim_{x \to 0} \frac{9}{\sqrt{1 - 81 x^{2}}}$

Step 5.2

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$9 lim_{x \to 0} \frac{1}{\sqrt{1 - 81 x^{2}}}$

Step 5.3

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

$9 \frac{lim_{x \to 0} 1}{lim_{x \to 0} \sqrt{1 - 81 x^{2}}}$

Step 5.4

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

$9 \frac{1}{lim_{x \to 0} \sqrt{1 - 81 x^{2}}}$

Step 5.5

Move the limit under the radical sign.

$9 \frac{1}{\sqrt{lim_{x \to 0} 1 - 81 x^{2}}}$

Step 5.6

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

$9 \frac{1}{\sqrt{lim_{x \to 0} 1 - lim_{x \to 0} 81 x^{2}}}$

Step 5.7

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

$9 \frac{1}{\sqrt{1 - lim_{x \to 0} 81 x^{2}}}$

Step 5.8

Move the term $81$ outside of the limit because it is constant with respect to $x$ .

$9 \frac{1}{\sqrt{1 - 81 lim_{x \to 0} x^{2}}}$

Step 5.9

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

$9 \frac{1}{\sqrt{1 - 81 {(lim_{x \to 0} x)}^{2}}}$

Step 6

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

$9 \frac{1}{\sqrt{1 - 81 \cdot 0^{2}}}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the denominator.

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Step 7.1.1

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

$9 \frac{1}{\sqrt{1 - 81 \cdot 0}}$

Step 7.1.2

Multiply $- 81$ by $0$ .

$9 \frac{1}{\sqrt{1 + 0}}$

Step 7.1.3

Add $1$ and $0$ .

$9 \frac{1}{\sqrt{1}}$

Step 7.1.4

Any root of $1$ is $1$ .

$9 (\frac{1}{1})$

Step 7.2

Cancel the common factor of $1$ .

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Step 7.2.1

Cancel the common factor.

$9 (\frac{1}{1})$

Step 7.2.2

Rewrite the expression.

$9 \cdot 1$

Step 7.3

Multiply $9$ by $1$ .

$9$