Calculus Examples

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

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.

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

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

Step 1.1.2.2

Evaluate the limit of $\arcsin (x)$ which is constant as $h$ approaches $0$ .

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

Step 1.1.2.3

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

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Step 1.1.2.3.1

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

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

Step 1.1.2.3.2

Add $x$ and $0$ .

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

Step 1.1.2.4

Subtract $\arcsin (x)$ from $\arcsin (x)$ .

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

Step 1.1.3

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

$\frac{0}{0}$

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

By the Sum Rule, the derivative of $\arcsin (x + h) - \arcsin (x)$ with respect to $h$ is $\frac{d}{d h} [\arcsin (x + h)] + \frac{d}{d h} [- \arcsin (x)]$ .

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

Step 1.3.3

Evaluate $\frac{d}{d h} [\arcsin (x + h)]$ .

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Step 1.3.3.1

Differentiate using the chain rule, which states that $\frac{d}{d h} [f (g (h))]$ is $f' (g (h)) g' (h)$ where $f (h) = \arcsin (h)$ and $g (h) = x + h$ .

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Step 1.3.3.1.1

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

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

Step 1.3.3.1.2

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

$lim_{h \to 0} \frac{\frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d h} [x + h] + \frac{d}{d h} [- \arcsin (x)]}{\frac{d}{d h} [h]}$

Step 1.3.3.1.3

Replace all occurrences of $u$ with $x + h$ .

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

Step 1.3.3.2

By the Sum Rule, the derivative of $x + h$ with respect to $h$ is $\frac{d}{d h} [x] + \frac{d}{d h} [h]$ .

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

Step 1.3.3.3

Since $x$ is constant with respect to $h$ , the derivative of $x$ with respect to $h$ is $0$ .

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

Step 1.3.3.4

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

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

Step 1.3.3.5

Add $0$ and $1$ .

$lim_{h \to 0} \frac{\frac{1}{\sqrt{1 - {(x + h)}^{2}}} \cdot 1 + \frac{d}{d h} [- \arcsin (x)]}{\frac{d}{d h} [h]}$

Step 1.3.3.6

Multiply $\frac{1}{\sqrt{1 - {(x + h)}^{2}}}$ by $1$ .

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

Step 1.3.4

Since $- \arcsin (x)$ is constant with respect to $h$ , the derivative of $- \arcsin (x)$ with respect to $h$ is $0$ .

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

Step 1.3.5

Add $\frac{1}{\sqrt{1 - {(x + h)}^{2}}}$ and $0$ .

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

Step 1.3.6

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

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

Multiply $\frac{1}{\sqrt{1 - {(x + h)}^{2}}}$ by $1$ .

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Move the limit under the radical sign.

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify the denominator.

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Step 4.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x + 0$ .

$\frac{1}{\sqrt{(1 + x + 0) (1 - (x + 0))}}$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Add $1 + x$ and $0$ .

$\frac{1}{\sqrt{(1 + x) (1 - (x + 0))}}$

Step 4.1.3.2

Add $x$ and $0$ .

$\frac{1}{\sqrt{(1 + x) (1 - x)}}$

Step 4.2

Multiply $\frac{1}{\sqrt{(1 + x) (1 - x)}}$ by $\frac{\sqrt{(1 + x) (1 - x)}}{\sqrt{(1 + x) (1 - x)}}$ .

$\frac{1}{\sqrt{(1 + x) (1 - x)}} \cdot \frac{\sqrt{(1 + x) (1 - x)}}{\sqrt{(1 + x) (1 - x)}}$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{1}{\sqrt{(1 + x) (1 - x)}}$ by $\frac{\sqrt{(1 + x) (1 - x)}}{\sqrt{(1 + x) (1 - x)}}$ .

$\frac{\sqrt{(1 + x) (1 - x)}}{\sqrt{(1 + x) (1 - x)} \sqrt{(1 + x) (1 - x)}}$

Step 4.3.2

Raise $\sqrt{(1 + x) (1 - x)}$ to the power of $1$ .

$\frac{\sqrt{(1 + x) (1 - x)}}{{\sqrt{(1 + x) (1 - x)}}^{1} \sqrt{(1 + x) (1 - x)}}$

Step 4.3.3

Raise $\sqrt{(1 + x) (1 - x)}$ to the power of $1$ .

$\frac{\sqrt{(1 + x) (1 - x)}}{{\sqrt{(1 + x) (1 - x)}}^{1} {\sqrt{(1 + x) (1 - x)}}^{1}}$

Step 4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{(1 + x) (1 - x)}}{{\sqrt{(1 + x) (1 - x)}}^{1 + 1}}$

Step 4.3.5

Add $1$ and $1$ .

$\frac{\sqrt{(1 + x) (1 - x)}}{{\sqrt{(1 + x) (1 - x)}}^{2}}$

Step 4.3.6

Rewrite ${\sqrt{(1 + x) (1 - x)}}^{2}$ as $(1 + x) (1 - x)$ .

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Step 4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(1 + x) (1 - x)}$ as ${((1 + x) (1 - x))}^{\frac{1}{2}}$ .

$\frac{\sqrt{(1 + x) (1 - x)}}{{({((1 + x) (1 - x))}^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{(1 + x) (1 - x)}}{{((1 + x) (1 - x))}^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{(1 + x) (1 - x)}}{{((1 + x) (1 - x))}^{\frac{2}{2}}}$

Step 4.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{(1 + x) (1 - x)}}{{((1 + x) (1 - x))}^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{(1 + x) (1 - x)}}{{((1 + x) (1 - x))}^{1}}$

Step 4.3.6.5

Simplify.

$\frac{\sqrt{(1 + x) (1 - x)}}{(1 + x) (1 - x)}$