Calculus Examples

$lim_{x \to \frac{\sqrt{3}}{2}} \frac{\arcsin (x) - \arcsin (\frac{\sqrt{3}}{2})}{x - \frac{\sqrt{3}}{2}}$

Step 1

Combine terms.

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Step 1.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to \frac{\sqrt{3}}{2}} \frac{\arcsin (x) - \arcsin (\frac{\sqrt{3}}{2})}{x \cdot \frac{2}{2} - \frac{\sqrt{3}}{2}}$

Step 1.2

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

$lim_{x \to \frac{\sqrt{3}}{2}} \frac{\arcsin (x) - \arcsin (\frac{\sqrt{3}}{2})}{\frac{x \cdot 2}{2} - \frac{\sqrt{3}}{2}}$

Step 1.3

Combine the numerators over the common denominator.

$lim_{x \to \frac{\sqrt{3}}{2}} \frac{\arcsin (x) - \arcsin (\frac{\sqrt{3}}{2})}{\frac{x \cdot 2 - \sqrt{3}}{2}}$

Step 2

Evaluate the limit.

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Step 2.1

Simplify the limit argument.

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Step 2.1.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to \frac{\sqrt{3}}{2}} (\arcsin (x) - \arcsin (\frac{\sqrt{3}}{2})) \frac{2}{x \cdot 2 - \sqrt{3}}$

Step 2.1.2

Multiply $\arcsin (x) - \arcsin (\frac{\sqrt{3}}{2})$ by $\frac{2}{x \cdot 2 - \sqrt{3}}$ .

$lim_{x \to \frac{\sqrt{3}}{2}} \frac{(\arcsin (x) - \arcsin (\frac{\sqrt{3}}{2})) \cdot 2}{x \cdot 2 - \sqrt{3}}$

Step 2.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{\arcsin (x) - \arcsin (\frac{\sqrt{3}}{2})}{x \cdot 2 - \sqrt{3}}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

Take the limit of the numerator and the limit of the denominator.

$2 \frac{lim_{x \to \frac{\sqrt{3}}{2}} \arcsin (x) - \arcsin (\frac{\sqrt{3}}{2})}{lim_{x \to \frac{\sqrt{3}}{2}} x \cdot 2 - \sqrt{3}}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{\sqrt{3}}{2}$ .

$2 \frac{lim_{x \to \frac{\sqrt{3}}{2}} \arcsin (x) - lim_{x \to \frac{\sqrt{3}}{2}} \arcsin (\frac{\sqrt{3}}{2})}{lim_{x \to \frac{\sqrt{3}}{2}} x \cdot 2 - \sqrt{3}}$

Step 3.1.2.2

Evaluate the limit of $\arcsin (\frac{\sqrt{3}}{2})$ which is constant as $x$ approaches $\frac{\sqrt{3}}{2}$ .

$2 \frac{lim_{x \to \frac{\sqrt{3}}{2}} \arcsin (x) - \arcsin (\frac{\sqrt{3}}{2})}{lim_{x \to \frac{\sqrt{3}}{2}} x \cdot 2 - \sqrt{3}}$

Step 3.1.2.3

Evaluate the limits by plugging in $\frac{\sqrt{3}}{2}$ for all occurrences of $x$ .

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Step 3.1.2.3.1

Evaluate the limit of $\arcsin (x)$ by plugging in $\frac{\sqrt{3}}{2}$ for $x$ .

$2 \frac{\arcsin (\frac{\sqrt{3}}{2}) - \arcsin (\frac{\sqrt{3}}{2})}{lim_{x \to \frac{\sqrt{3}}{2}} x \cdot 2 - \sqrt{3}}$

Step 3.1.2.3.2

The exact value of $\arcsin (\frac{\sqrt{3}}{2})$ is $\frac{π}{3}$ .

$2 \frac{\frac{π}{3} - \arcsin (\frac{\sqrt{3}}{2})}{lim_{x \to \frac{\sqrt{3}}{2}} x \cdot 2 - \sqrt{3}}$

Step 3.1.2.4

Simplify the answer.

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Step 3.1.2.4.1

The exact value of $\arcsin (\frac{\sqrt{3}}{2})$ is $\frac{π}{3}$ .

$2 \frac{\frac{π}{3} - \frac{π}{3}}{lim_{x \to \frac{\sqrt{3}}{2}} x \cdot 2 - \sqrt{3}}$

Step 3.1.2.4.2

Combine the numerators over the common denominator.

$2 \frac{\frac{π - π}{3}}{lim_{x \to \frac{\sqrt{3}}{2}} x \cdot 2 - \sqrt{3}}$

Step 3.1.2.4.3

Subtract $π$ from $π$ .

$2 \frac{\frac{0}{3}}{lim_{x \to \frac{\sqrt{3}}{2}} x \cdot 2 - \sqrt{3}}$

Step 3.1.2.4.4

Divide $0$ by $3$ .

$2 \frac{0}{lim_{x \to \frac{\sqrt{3}}{2}} x \cdot 2 - \sqrt{3}}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Evaluate the limit.

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Step 3.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{\sqrt{3}}{2}$ .

$2 \frac{0}{lim_{x \to \frac{\sqrt{3}}{2}} x \cdot 2 - lim_{x \to \frac{\sqrt{3}}{2}} \sqrt{3}}$

Step 3.1.3.1.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$2 \frac{0}{2 lim_{x \to \frac{\sqrt{3}}{2}} x - lim_{x \to \frac{\sqrt{3}}{2}} \sqrt{3}}$

Step 3.1.3.1.3

Evaluate the limit of $\sqrt{3}$ which is constant as $x$ approaches $\frac{\sqrt{3}}{2}$ .

$2 \frac{0}{2 lim_{x \to \frac{\sqrt{3}}{2}} x - \sqrt{3}}$

Step 3.1.3.2

Evaluate the limit of $x$ by plugging in $\frac{\sqrt{3}}{2}$ for $x$ .

$2 \frac{0}{2 \frac{\sqrt{3}}{2} - \sqrt{3}}$

Step 3.1.3.3

Simplify the answer.

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Step 3.1.3.3.1

Cancel the common factor of $2$ .

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Step 3.1.3.3.1.1

Cancel the common factor.

$2 \frac{0}{2 \frac{\sqrt{3}}{2} - \sqrt{3}}$

Step 3.1.3.3.1.2

Rewrite the expression.

$2 \frac{0}{\sqrt{3} - \sqrt{3}}$

Step 3.1.3.3.2

Subtract $\sqrt{3}$ from $\sqrt{3}$ .

$2 (\frac{0}{0})$

Step 3.1.3.3.3

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

Undefined

$2 (\frac{0}{0})$

Step 3.1.3.4

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

Undefined

$2 (\frac{0}{0})$

Step 3.1.4

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

Undefined

$2 (\frac{0}{0})$

Step 3.2

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.

$lim_{x \to \frac{\sqrt{3}}{2}} \frac{\arcsin (x) - \arcsin (\frac{\sqrt{3}}{2})}{x \cdot 2 - \sqrt{3}} = lim_{x \to \frac{\sqrt{3}}{2}} \frac{\frac{d}{d x} [\arcsin (x) - \arcsin (\frac{\sqrt{3}}{2})]}{\frac{d}{d x} [x \cdot 2 - \sqrt{3}]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{\frac{d}{d x} [\arcsin (x) - \arcsin (\frac{\sqrt{3}}{2})]}{\frac{d}{d x} [x \cdot 2 - \sqrt{3}]}$

Step 3.3.2

The exact value of $\arcsin (\frac{\sqrt{3}}{2})$ is $\frac{π}{3}$ .

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{\frac{d}{d x} [\arcsin (x) - \frac{π}{3}]}{\frac{d}{d x} [x \cdot 2 - \sqrt{3}]}$

Step 3.3.3

By the Sum Rule, the derivative of $\arcsin (x) - \frac{π}{3}$ with respect to $x$ is $\frac{d}{d x} [\arcsin (x)] + \frac{d}{d x} [- \frac{π}{3}]$ .

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{\frac{d}{d x} [\arcsin (x)] + \frac{d}{d x} [- \frac{π}{3}]}{\frac{d}{d x} [x \cdot 2 - \sqrt{3}]}$

Step 3.3.4

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

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{\frac{1}{\sqrt{1 - x^{2}}} + \frac{d}{d x} [- \frac{π}{3}]}{\frac{d}{d x} [x \cdot 2 - \sqrt{3}]}$

Step 3.3.5

Since $- \frac{π}{3}$ is constant with respect to $x$ , the derivative of $- \frac{π}{3}$ with respect to $x$ is $0$ .

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{\frac{1}{\sqrt{1 - x^{2}}} + 0}{\frac{d}{d x} [x \cdot 2 - \sqrt{3}]}$

Step 3.3.6

Add $\frac{1}{\sqrt{1 - x^{2}}}$ and $0$ .

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{\frac{1}{\sqrt{1 - x^{2}}}}{\frac{d}{d x} [x \cdot 2 - \sqrt{3}]}$

Step 3.3.7

By the Sum Rule, the derivative of $x \cdot 2 - \sqrt{3}$ with respect to $x$ is $\frac{d}{d x} [x \cdot 2] + \frac{d}{d x} [- \sqrt{3}]$ .

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{\frac{1}{\sqrt{1 - x^{2}}}}{\frac{d}{d x} [x \cdot 2] + \frac{d}{d x} [- \sqrt{3}]}$

Step 3.3.8

Evaluate $\frac{d}{d x} [x \cdot 2]$ .

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Step 3.3.8.1

Move $2$ to the left of $x$ .

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{\frac{1}{\sqrt{1 - x^{2}}}}{\frac{d}{d x} [2 \cdot x] + \frac{d}{d x} [- \sqrt{3}]}$

Step 3.3.8.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{\frac{1}{\sqrt{1 - x^{2}}}}{2 \frac{d}{d x} [x] + \frac{d}{d x} [- \sqrt{3}]}$

Step 3.3.8.3

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

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{\frac{1}{\sqrt{1 - x^{2}}}}{2 \cdot 1 + \frac{d}{d x} [- \sqrt{3}]}$

Step 3.3.8.4

Multiply $2$ by $1$ .

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{\frac{1}{\sqrt{1 - x^{2}}}}{2 + \frac{d}{d x} [- \sqrt{3}]}$

Step 3.3.9

Since $- \sqrt{3}$ is constant with respect to $x$ , the derivative of $- \sqrt{3}$ with respect to $x$ is $0$ .

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{\frac{1}{\sqrt{1 - x^{2}}}}{2 + 0}$

Step 3.3.10

Add $2$ and $0$ .

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{\frac{1}{\sqrt{1 - x^{2}}}}{2}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1 - x^{2}}} \cdot \frac{1}{2}$

Step 3.5

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

$2 lim_{x \to \frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1 - x^{2}} \cdot 2}$

Step 4

Evaluate the limit.

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Step 4.1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$2 (\frac{1}{2}) lim_{x \to \frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1 - x^{2}}}$

Step 4.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{\sqrt{3}}{2}$ .

$2 (\frac{1}{2}) \frac{lim_{x \to \frac{\sqrt{3}}{2}} 1}{lim_{x \to \frac{\sqrt{3}}{2}} \sqrt{1 - x^{2}}}$

Step 4.3

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{\sqrt{3}}{2}$ .

$2 (\frac{1}{2}) \frac{1}{lim_{x \to \frac{\sqrt{3}}{2}} \sqrt{1 - x^{2}}}$

Step 4.4

Move the limit under the radical sign.

$2 (\frac{1}{2}) \frac{1}{\sqrt{lim_{x \to \frac{\sqrt{3}}{2}} 1 - x^{2}}}$

Step 4.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{\sqrt{3}}{2}$ .

$2 (\frac{1}{2}) \frac{1}{\sqrt{lim_{x \to \frac{\sqrt{3}}{2}} 1 - lim_{x \to \frac{\sqrt{3}}{2}} x^{2}}}$

Step 4.6

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{\sqrt{3}}{2}$ .

$2 (\frac{1}{2}) \frac{1}{\sqrt{1 - lim_{x \to \frac{\sqrt{3}}{2}} x^{2}}}$

Step 4.7

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

$2 (\frac{1}{2}) \frac{1}{\sqrt{1 - {(lim_{x \to \frac{\sqrt{3}}{2}} x)}^{2}}}$

Step 5

Evaluate the limit of $x$ by plugging in $\frac{\sqrt{3}}{2}$ for $x$ .

$2 (\frac{1}{2}) \frac{1}{\sqrt{1 - {(\frac{\sqrt{3}}{2})}^{2}}}$

Step 6

Simplify the answer.

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Step 6.1

Cancel the common factor of $2$ .

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Step 6.1.1

Cancel the common factor.

$2 (\frac{1}{2}) \frac{1}{\sqrt{1 - {(\frac{\sqrt{3}}{2})}^{2}}}$

Step 6.1.2

Rewrite the expression.

$1 \frac{1}{\sqrt{1 - {(\frac{\sqrt{3}}{2})}^{2}}}$

Step 6.2

Multiply $\frac{1}{\sqrt{1 - {(\frac{\sqrt{3}}{2})}^{2}}}$ by $1$ .

$\frac{1}{\sqrt{1 - {(\frac{\sqrt{3}}{2})}^{2}}}$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$\frac{1}{\sqrt{1 - \frac{{\sqrt{3}}^{2}}{2^{2}}}}$

Step 6.3.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{1}{\sqrt{1 - \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}}}$

Step 6.3.2.2

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

$\frac{1}{\sqrt{1 - \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}}}$

Step 6.3.2.3

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

$\frac{1}{\sqrt{1 - \frac{3^{\frac{2}{2}}}{2^{2}}}}$

Step 6.3.2.4

Cancel the common factor of $2$ .

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Step 6.3.2.4.1

Cancel the common factor.

$\frac{1}{\sqrt{1 - \frac{3^{\frac{2}{2}}}{2^{2}}}}$

Step 6.3.2.4.2

Rewrite the expression.

$\frac{1}{\sqrt{1 - \frac{3^{1}}{2^{2}}}}$

Step 6.3.2.5

Evaluate the exponent.

$\frac{1}{\sqrt{1 - \frac{3}{2^{2}}}}$

Step 6.3.3

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

$\frac{1}{\sqrt{1 - \frac{3}{4}}}$

Step 6.3.4

Write $1$ as a fraction with a common denominator.

$\frac{1}{\sqrt{\frac{4}{4} - \frac{3}{4}}}$

Step 6.3.5

Combine the numerators over the common denominator.

$\frac{1}{\sqrt{\frac{4 - 3}{4}}}$

Step 6.3.6

Subtract $3$ from $4$ .

$\frac{1}{\sqrt{\frac{1}{4}}}$

Step 6.3.7

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$\frac{1}{\frac{\sqrt{1}}{\sqrt{4}}}$

Step 6.3.8

Any root of $1$ is $1$ .

$\frac{1}{\frac{1}{\sqrt{4}}}$

Step 6.3.9

Simplify the denominator.

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Step 6.3.9.1

Rewrite $4$ as $2^{2}$ .

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

Step 6.3.9.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{\frac{1}{2}}$

Step 6.4

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Step 6.5

Multiply $2$ by $1$ .

$2$