# Limits by rationalizing Calculator

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### Difficult Problems

1

Example

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

Applying rationalisation

$\lim_{x\to0}\left(\frac{\sqrt{5}+\sqrt{x+5}}{\sqrt{5}+\sqrt{x+5}}\cdot\frac{\sqrt{x+5}-\sqrt{5}}{x}\right)$
3

Multiplying fractions

$\lim_{x\to0}\left(\frac{\left(\sqrt{5}+\sqrt{x+5}\right)\left(\sqrt{x+5}-\sqrt{5}\right)}{x\left(\sqrt{5}+\sqrt{x+5}\right)}\right)$
4

By multiplying conjugated binomials you get a difference of squares: $(a+b)(a-b)=a^2-b^2$

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

Applying the power of a power property

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

Subtract the values $5$ and $-5$

$\lim_{x\to0}\left(\frac{x}{x\left(\sqrt{5}+\sqrt{x+5}\right)}\right)$
7

Simplifying the fraction by $x$

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

Evaluating the limit when $x$ tends to $0$

$\frac{1}{2.2361+\sqrt{0+5}}$
9

Simplifying

$\frac{36}{161}$