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# Limits by Rationalizing Calculator

## Get detailed solutions to your math problems with our Limits by Rationalizing step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

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###  Difficult Problems

1

Here, we show you a step-by-step solved example of limits by rationalizing. This solution was automatically generated by our smart calculator:

$\lim_{x\to0}\left(\frac{\sqrt{5+x}-\sqrt{5}}{x}\right)$
2

Simplifying

$\lim_{x\to0}\left(\frac{\sqrt{5+x}-\sqrt{5}}{x}\right)$

Plug in the value $0$ into the limit

$\frac{\sqrt{5+0}-\sqrt{5}}{0}$

Add the values $5$ and $0$

$\frac{\sqrt{5}-2.2361}{0}$

Calculate the power $\sqrt{5}$

$\frac{2.2361-2.2361}{0}$

Subtract the values $2.2361$ and $-2.2361$

$\frac{0}{0}$
3

If we directly evaluate the limit $\lim_{x\to 0}\left(\frac{\sqrt{5+x}-\sqrt{5}}{x}\right)$ as $x$ tends to $0$, we can see that it gives us an indeterminate form

$\frac{0}{0}$
4

We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately

$\lim_{x\to 0}\left(\frac{\frac{d}{dx}\left(\sqrt{5+x}-\sqrt{5}\right)}{\frac{d}{dx}\left(x\right)}\right)$

Find the derivative of the numerator

$\frac{d}{dx}\left(\sqrt{5+x}-\sqrt{5}\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(\sqrt{5+x}\right)+\frac{d}{dx}\left(-\sqrt{5}\right)$

The derivative of the constant function ($-\sqrt{5}$) is equal to zero

$\frac{d}{dx}\left(\sqrt{5+x}\right)$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{1}{2}\left(5+x\right)^{-\frac{1}{2}}\frac{d}{dx}\left(5+x\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{1}{2}\left(5+x\right)^{-\frac{1}{2}}\left(\frac{d}{dx}\left(5\right)+\frac{d}{dx}\left(x\right)\right)$

The derivative of the constant function ($5$) is equal to zero

$\frac{1}{2}\left(5+x\right)^{-\frac{1}{2}}\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$\frac{1}{2}\left(5+x\right)^{-\frac{1}{2}}$

Find the derivative of the denominator

$\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$1$

Any expression divided by one ($1$) is equal to that same expression

$\lim_{x\to0}\left(\frac{1}{2}\left(5+x\right)^{-\frac{1}{2}}\right)$
5

After deriving both the numerator and denominator, the limit results in

$\lim_{x\to0}\left(\frac{1}{2}\left(5+x\right)^{-\frac{1}{2}}\right)$
6

The limit of the product of a function and a constant is equal to the limit of the function, times the constant: $\displaystyle \lim_{t\to 0}{\left(at\right)}=a\cdot\lim_{t\to 0}{\left(t\right)}$

$\frac{1}{2}\lim_{x\to0}\left(\left(5+x\right)^{-\frac{1}{2}}\right)$
7

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{1}{2}\lim_{x\to0}\left(\frac{1}{\sqrt{5+x}}\right)$
8

Evaluate the limit $\lim_{x\to0}\left(\frac{1}{\sqrt{5+x}}\right)$ by replacing all occurrences of $x$ by $0$

$\frac{1}{2}\cdot \left(\frac{1}{\sqrt{5+0}}\right)$
9

Add the values $5$ and $0$

$\frac{1}{2}\cdot \left(\frac{1}{\sqrt{5}}\right)$
10

Calculate the power $\sqrt{5}$

$\frac{1}{2}\cdot \left(\frac{1}{\sqrt{5}}\right)$
11

Divide $1$ by $\sqrt{5}$

$\frac{1}{2}\frac{\sqrt{5}}{5}$
12

Multiply $\frac{1}{2}$ times $\frac{\sqrt{5}}{5}$

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

##  Final answer to the problem

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

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