Step-by-step Solution

Evaluate the limit of $\frac{\sqrt{5+x}-1\cdot \sqrt{5}}{x}$ as $x$ approaches 0

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Step-by-step explanation

Problem to solve:

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

Learn how to solve limits problems step by step online.

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

Unlock this full step-by-step solution!

Learn how to solve limits problems step by step online. Evaluate the limit of ((5+x)^0.5-5^0.5)/x as x approaches 0. Applying rationalisation. Multiplying fractions \frac{\sqrt{5+x}-\frac{5}{\sqrt{5}}}{x} \times \frac{\sqrt{5+x}+\frac{5}{\sqrt{5}}}{\sqrt{5+x}+\frac{5}{\sqrt{5}}}. Solve the product of difference of squares \left(\sqrt{5+x}-\frac{5}{\sqrt{5}}\right)\left(\sqrt{5+x}+\frac{5}{\sqrt{5}}\right). Simplify the fraction \frac{x}{x\left(\sqrt{5+x}+\frac{5}{\sqrt{5}}\right)} by x.

Final Answer

$\frac{1}{\sqrt{20}}$$\,\,\left(\approx 0.22360679774997896\right)$
$\lim_{x\to0}\left(\frac{\sqrt{5+x}-\sqrt{5}}{x}\right)$

Main topic:

Limits

Time to solve it:

~ 0.03 s (SnapXam)