Evaluate limit as x approaches two hundred twenty-five of ( square root of x minus fifteen) divide(x minus two hundred twenty-five)

$lim_{x \to 225} \frac{\sqrt{x} - 15}{x - 225}$

Take the limit of each term.

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Apply L'Hospital's rule.

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

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

$\frac{lim_{x \to 225} \sqrt{x} - 15}{lim_{x \to 225} x - 225}$

Evaluate the limit of the numerator.

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First, multiply the argument of the limit by the conjugate.

$\frac{lim_{x \to 225} (\sqrt{x} - 15) (\sqrt{x} + 15)}{lim_{x \to 225} x - 225}$

Expand $(\sqrt{x} - 15) (\sqrt{x} + 15)$ using the FOIL Method.

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Use the distributive law.

$\frac{lim_{x \to 225} \sqrt{x} (\sqrt{x} + 15) - 15 (\sqrt{x} + 15)}{lim_{x \to 225} x - 225}$

Apply the distributive property.

$\frac{lim_{x \to 225} \sqrt{x} \sqrt{x} + \sqrt{x} \cdot 15 - 15 (\sqrt{x} + 15)}{lim_{x \to 225} x - 225}$

Apply the distributive property.

$\frac{lim_{x \to 225} \sqrt{x} \sqrt{x} + \sqrt{x} \cdot 15 - 15 \sqrt{x} - 15 \cdot 15}{lim_{x \to 225} x - 225}$

Add the opposite terms in $\sqrt{x} \sqrt{x} + \sqrt{x} \cdot 15 - 15 \sqrt{x} - 15 \cdot 15$ .

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Shift the factors in the terms $\sqrt{x} \cdot 15$ and $- 15 \sqrt{x}$ .

$\frac{lim_{x \to 225} \sqrt{x} \sqrt{x} + 15 \sqrt{x} - 15 \sqrt{x} - 15 \cdot 15}{lim_{x \to 225} x - 225}$

Subtract $15 \sqrt{x}$ from $15 \sqrt{x}$ .

$\frac{lim_{x \to 225} \sqrt{x} \sqrt{x} + 0 - 15 \cdot 15}{lim_{x \to 225} x - 225}$

Add $\sqrt{x} \sqrt{x}$ and $0$ .

$\frac{lim_{x \to 225} \sqrt{x} \sqrt{x} - 15 \cdot 15}{lim_{x \to 225} x - 225}$

Clarify each term.

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Multiply $\sqrt{x} \sqrt{x}$ .

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Raise $\sqrt{x}$ to the power of $1$ .

$\frac{lim_{x \to 225} {\sqrt{x}}^{1} \sqrt{x} - 15 \cdot 15}{lim_{x \to 225} x - 225}$

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{lim_{x \to 225} {\sqrt{x}}^{1} {\sqrt{x}}^{1} - 15 \cdot 15}{lim_{x \to 225} x - 225}$

Apply the power rule $a^{m} a^{n} = a^{m + n}$ to Add exponents.

$\frac{lim_{x \to 225} {\sqrt{x}}^{1 + 1} - 15 \cdot 15}{lim_{x \to 225} x - 225}$

Add $1$ and $1$ .

$\frac{lim_{x \to 225} {\sqrt{x}}^{2} - 15 \cdot 15}{lim_{x \to 225} x - 225}$

Should be rewritten ${\sqrt{x}}^{2}$ as $x$ .

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

$\frac{lim_{x \to 225} {(x^{\frac{1}{2}})}^{2} - 15 \cdot 15}{lim_{x \to 225} x - 225}$

Use the power rule when multiplying the exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{lim_{x \to 225} x^{\frac{1}{2} \cdot 2} - 15 \cdot 15}{lim_{x \to 225} x - 225}$

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

$\frac{lim_{x \to 225} x^{\frac{2}{2}} - 15 \cdot 15}{lim_{x \to 225} x - 225}$

The common factor should be canceled of $2$ .

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Cancel the common factor.

$\frac{lim_{x \to 225} x^{\frac{2}{2}} - 15 \cdot 15}{lim_{x \to 225} x - 225}$

First, divide $1$ by $1$ .

$\frac{lim_{x \to 225} x^{1} - 15 \cdot 15}{lim_{x \to 225} x - 225}$

Simplify.

$\frac{lim_{x \to 225} x - 15 \cdot 15}{lim_{x \to 225} x - 225}$

Multiply $- 15$ by $15$ .

$\frac{lim_{x \to 225} x - 225}{lim_{x \to 225} x - 225}$

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $225$ .

$\frac{lim_{x \to 225} x - lim_{x \to 225} 225}{lim_{x \to 225} x - 225}$

Evaluate the limits by plugging in $225$ for all occurrences of $x$ .

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Evaluate the limit of $x$ by plugging in $225$ for $x$ .

$\frac{225 - lim_{x \to 225} 225}{lim_{x \to 225} x - 225}$

Evaluate the limit of $225$ which is constant as $x$ approaches $225$ .

$\frac{225 - 225}{lim_{x \to 225} x - 225}$

Subtract $225$ from $225$ .

$\frac{0}{lim_{x \to 225} x - 225}$

Evaluate the limit of the denominator.

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Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $225$ .

$\frac{0}{lim_{x \to 225} x - lim_{x \to 225} 225}$

Evaluate the limits by plugging in $225$ for all occurrences of $x$ .

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Evaluate the limit of $x$ by plugging in $225$ for $x$ .

$\frac{0}{225 - lim_{x \to 225} 225}$

Evaluate the limit of $225$ which is constant as $x$ approaches $225$ .

$\frac{0}{225 - 225}$

Subtract $225$ from $225$ .

$\frac{0}{0}$

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

Undefined

$\frac{0}{0}$

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

Undefined

$\frac{0}{0}$

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.

$lim_{x \to 225} \frac{\sqrt{x} - 15}{x - 225} = lim_{x \to 225} \frac{\frac{d}{d x} [\sqrt{x} - 15]}{\frac{d}{d x} [x - 225]}$

Find the derivative of the numerator and denominator.

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Differentiate the numerator and denominator.

$lim_{x \to 225} \frac{\frac{d}{d x} [\sqrt{x} - 15]}{\frac{d}{d x} [x - 225]}$

By the Sum Rule, the derivative of $\sqrt{x} - 15$ with respect to $x$ is $\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [- 15]$ .

$lim_{x \to 225} \frac{\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x - 225]}$

Evaluate $\frac{d}{d x} [\sqrt{x}]$ .

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

$lim_{x \to 225} \frac{\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x - 225]}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{x \to 225} \frac{\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x - 225]}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 225} \frac{\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x - 225]}$

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

$lim_{x \to 225} \frac{\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x - 225]}$

Add numerators over the common denominator.

$lim_{x \to 225} \frac{\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x - 225]}$

Clarify the numerator.

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Multiply $- 1$ by $2$ .

$lim_{x \to 225} \frac{\frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x - 225]}$

Subtract $2$ from $1$ .

$lim_{x \to 225} \frac{\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x - 225]}$

Put the negative in front of the fraction.

$lim_{x \to 225} \frac{\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x - 225]}$

Since $- 15$ is constant with respect to $x$ , the derivative of $- 15$ with respect to $x$ is $0$ .

$lim_{x \to 225} \frac{\frac{1}{2} x^{- \frac{1}{2}} + 0}{\frac{d}{d x} [x - 225]}$

Reformulate the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 225} \frac{\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}} + 0}{\frac{d}{d x} [x - 225]}$

Combine terms.

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Multiply $\frac{1}{2}$ and $\frac{1}{x^{\frac{1}{2}}}$ .

$lim_{x \to 225} \frac{\frac{1}{2 x^{\frac{1}{2}}} + 0}{\frac{d}{d x} [x - 225]}$

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

$lim_{x \to 225} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [x - 225]}$

By the Sum Rule, the derivative of $x - 225$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 225]$ .

$lim_{x \to 225} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [x] + \frac{d}{d x} [- 225]}$

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

$lim_{x \to 225} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{1 + \frac{d}{d x} [- 225]}$

Since $- 225$ is constant with respect to $x$ , the derivative of $- 225$ with respect to $x$ is $0$ .

$lim_{x \to 225} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{1 + 0}$

Add $1$ and $0$ .

$lim_{x \to 225} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{1}$

Multiply the numerator by the denominator's reciprocal.

$lim_{x \to 225} \frac{1}{2 x^{\frac{1}{2}}} \cdot 1$

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$lim_{x \to 225} \frac{1}{2 \sqrt{x}} \cdot 1$

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

$lim_{x \to 225} \frac{1}{2 \sqrt{x}}$

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

$\frac{1}{2} lim_{x \to 225} \frac{1}{\sqrt{x}}$

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $225$ .

$\frac{1}{2} \cdot \frac{lim_{x \to 225} 1}{lim_{x \to 225} \sqrt{x}}$

Move the limit under the radical sign.

$\frac{1}{2} \cdot \frac{lim_{x \to 225} 1}{\sqrt{lim_{x \to 225} x}}$

Evaluate the limits by plugging in $225$ for all occurrences of $x$ .

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Evaluate the limit of $1$ which is constant as $x$ approaches $225$ .

$\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 225} x}}$

Evaluate the limit of $x$ by plugging in $225$ for $x$ .

$\frac{1}{2} \cdot \frac{1}{\sqrt{225}}$

Clarify the answer.

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Clarify the denominator.

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Rewrite $225$ as $15^{2}$ .

$\frac{1}{2} \cdot \frac{1}{\sqrt{15^{2}}}$

Withdraw terms from under the radical, assuming positive real numbers.

$\frac{1}{2} \cdot \frac{1}{15}$

Multiply $\frac{1}{2} \cdot \frac{1}{15}$ .

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Multiply $\frac{1}{2}$ and $\frac{1}{15}$ .

$\frac{1}{2 \cdot 15}$

First, multiply $2$ by $15$ .

$\frac{1}{30}$

The result can be displayed in a variety of ways.

Exact Form:

$\frac{1}{30}$

Decimal Form:

$0.0\overline{3}$

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