Calculus Examples

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

Step 1

Apply L'Hospital's rule.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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 1.1.2

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

Tap for more steps...

Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.3

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

$\frac{0}{0}$

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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 1.3.3

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 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

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

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

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

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 2.2

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

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

Step 2.3

Move the limit under the radical sign.

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

Step 2.4

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 2.5

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

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

Step 2.6

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 3

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

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

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Simplify the denominator.

Tap for more steps...

Step 4.1.1

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

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

Step 4.1.2

Multiply $- 1$ by $0$ .

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

Step 4.1.3

Add $1$ and $0$ .

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

Step 4.1.4

Any root of $1$ is $1$ .

$\frac{1}{1}$

Step 4.2

Divide $1$ by $1$ .

$1$