Simplify limit as x approaches 0 of ( square root of x+5- square root of 5)/x

$lim_{x \to 0} \frac{\sqrt{x + 5} - \sqrt{5}}{x}$

Step 1

Apply L\'Hospital\'s rule.

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

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

$\frac{lim_{x \to 0} \sqrt{x + 5} - \sqrt{5}}{lim_{x \to 0} x}$

Evaluate the limit of the numerator.

Evaluate the limit.

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

$\frac{lim_{x \to 0} \sqrt{x + 5} - lim_{x \to 0} \sqrt{5}}{lim_{x \to 0} x}$

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 0} x + 5} - lim_{x \to 0} \sqrt{5}}{lim_{x \to 0} x}$

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

$\frac{\sqrt{lim_{x \to 0} x + lim_{x \to 0} 5} - lim_{x \to 0} \sqrt{5}}{lim_{x \to 0} x}$

Evaluate the limit of $5$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt{lim_{x \to 0} x + 5} - lim_{x \to 0} \sqrt{5}}{lim_{x \to 0} x}$

Evaluate the limit of $\sqrt{5}$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt{lim_{x \to 0} x + 5} - \sqrt{5}}{lim_{x \to 0} x}$

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

$\frac{\sqrt{0 + 5} - \sqrt{5}}{lim_{x \to 0} x}$

Simplify the answer.

Add $0$ and $5$ .

$\frac{\sqrt{5} - \sqrt{5}}{lim_{x \to 0} x}$

Subtract $\sqrt{5}$ from $\sqrt{5}$ .

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

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

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{\sqrt{x + 5} - \sqrt{5}}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\sqrt{x + 5} - \sqrt{5}]}{\frac{d}{d x} [x]}$

Find the derivative of the numerator and denominator.

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\sqrt{x + 5} - \sqrt{5}]}{\frac{d}{d x} [x]}$

By the Sum Rule, the derivative of $\sqrt{x + 5} - \sqrt{5}$ with respect to $x$ is $\frac{d}{d x} [\sqrt{x + 5}] + \frac{d}{d x} [- \sqrt{5}]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [\sqrt{x + 5}] + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

Evaluate $\frac{d}{d x} [\sqrt{x + 5}]$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 5}$ as ${(x + 5)}^{\frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [{(x + 5)}^{\frac{1}{2}}] + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x + 5$ .

To apply the Chain Rule, set $u$ as $x + 5$ .

$lim_{x \to 0} \frac{\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x + 5] + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

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

$lim_{x \to 0} \frac{\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x + 5] + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

Replace all occurrences of $u$ with $x + 5$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 5)}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 5] + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

By the Sum Rule, the derivative of $x + 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 5)}^{\frac{1}{2} - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [5]) + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

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

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 5)}^{\frac{1}{2} - 1} (1 + \frac{d}{d x} [5]) + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 5)}^{\frac{1}{2} - 1} (1 + 0) + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

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

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 5)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (1 + 0) + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

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

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 5)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (1 + 0) + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 5)}^{\frac{1 - 1 \cdot 2}{2}} (1 + 0) + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 5)}^{\frac{1 - 2}{2}} (1 + 0) + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

Subtract $2$ from $1$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 5)}^{\frac{- 1}{2}} (1 + 0) + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 5)}^{- \frac{1}{2}} (1 + 0) + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

Add $1$ and $0$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 5)}^{- \frac{1}{2}} \cdot 1 + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

Combine $\frac{1}{2}$ and ${(x + 5)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{{(x + 5)}^{- \frac{1}{2}}}{2} \cdot 1 + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

Multiply $\frac{{(x + 5)}^{- \frac{1}{2}}}{2}$ by $1$ .

$lim_{x \to 0} \frac{\frac{{(x + 5)}^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

Move ${(x + 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 5)}^{\frac{1}{2}}} + \frac{d}{d x} [- \sqrt{5}]}{\frac{d}{d x} [x]}$

Since $- \sqrt{5}$ is constant with respect to $x$ , the derivative of $- \sqrt{5}$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 5)}^{\frac{1}{2}}} + 0}{\frac{d}{d x} [x]}$

Add $\frac{1}{2 {(x + 5)}^{\frac{1}{2}}}$ and $0$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 5)}^{\frac{1}{2}}}}{\frac{d}{d x} [x]}$

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

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 5)}^{\frac{1}{2}}}}{1}$

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{1}{2 {(x + 5)}^{\frac{1}{2}}} \cdot 1$

Rewrite ${(x + 5)}^{\frac{1}{2}}$ as $\sqrt{x + 5}$ .

$lim_{x \to 0} \frac{1}{2 \sqrt{x + 5}} \cdot 1$

Multiply $\frac{1}{2 \sqrt{x + 5}}$ by $1$ .

$lim_{x \to 0} \frac{1}{2 \sqrt{x + 5}}$

Step 2

Evaluate the limit.

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

$\frac{1}{2} lim_{x \to 0} \frac{1}{\sqrt{x + 5}}$

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

$\frac{1}{2} \cdot \frac{lim_{x \to 0} 1}{lim_{x \to 0} \sqrt{x + 5}}$

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

$\frac{1}{2} \cdot \frac{1}{lim_{x \to 0} \sqrt{x + 5}}$

Move the limit under the radical sign.

$\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 0} x + 5}}$

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

$\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 0} x + lim_{x \to 0} 5}}$

Evaluate the limit of $5$ which is constant as $x$ approaches $0$ .

$\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 0} x + 5}}$

Step 3

Simplify the denominator.

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

$\frac{1}{2} \cdot \frac{1}{\sqrt{0 + 5}}$

Add $0$ and $5$ .

$\frac{1}{2} \cdot \frac{1}{\sqrt{5}}$

Step 4

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{1}{2} (\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 5

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{1}{2} \cdot \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{1}{2} \cdot \frac{\sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{1}{2} \cdot \frac{\sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

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

$\frac{1}{2} \cdot \frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{1}{2} \cdot \frac{\sqrt{5}}{{\sqrt{5}}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{1}{2} \cdot \frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

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

$\frac{1}{2} \cdot \frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

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

$\frac{1}{2} \cdot \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{1}{2} \cdot \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{1}{2} \cdot \frac{\sqrt{5}}{5^{1}}$

Evaluate the exponent.

$\frac{1}{2} \cdot \frac{\sqrt{5}}{5}$

Step 6

Multiply $\frac{1}{2} \cdot \frac{\sqrt{5}}{5}$ .

Multiply $\frac{1}{2}$ by $\frac{\sqrt{5}}{5}$ .

$\frac{\sqrt{5}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$\frac{\sqrt{5}}{10}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{5}}{10}$

Decimal Form:

$0.22360679 \dots$

Simplify limit as x approaches 0 of ( square root of x+5- square root of 5)/x

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