Calculus Examples

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

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.

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

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 $\arcsin (x)$ by plugging in $0$ for $x$ .

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

Step 1.2.2

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Evaluate the limit.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Move the limit under the radical sign.

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

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

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

Step 6

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

$\frac{1}{\sqrt{1 - 0^{2}}}$

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 $0$ .

$\frac{1}{\sqrt{1 - 0}}$

Step 7.1.2

Multiply $- 1$ by $0$ .

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

Step 7.1.3

Add $1$ and $0$ .

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

Step 7.1.4

Any root of $1$ is $1$ .

$\frac{1}{1}$

Step 7.2

Divide $1$ by $1$ .

$1$