Here, we show you a step-by-step solved example of limits by rationalizing. This solution was automatically generated by our smart calculator:
Applying rationalisation
Multiply and simplify the expression within the limit
Multiplying fractions $\frac{\sqrt{5+x}-\sqrt{5}}{x} \times \frac{\sqrt{5+x}+\sqrt{5}}{\sqrt{5+x}+\sqrt{5}}$
The first term ($a$) is $\sqrt{5+x}$.
The second term ($b$) is $\sqrt{5}$.
Solve the product of difference of squares $\left(\sqrt{5+x}-\sqrt{5}\right)\left(\sqrt{5+x}+\sqrt{5}\right)$
Cancel exponents $\frac{1}{2}$ and $2$
Cancel exponents $\frac{1}{2}$ and $2$
Multiply and simplify the expression within the limit
Subtract the values $5$ and $-5$
Simplify the fraction $\frac{x}{x\left(\sqrt{5+x}+\sqrt{5}\right)}$ by $x$
Evaluate the limit $\lim_{x\to0}\left(\frac{1}{\sqrt{5+x}+\sqrt{5}}\right)$ by replacing all occurrences of $x$ by $0$
Add the values $5$ and $0$
Combining like terms $\sqrt{5}$ and $\sqrt{5}$
Evaluate the limit $\lim_{x\to0}\left(\frac{1}{\sqrt{5+x}+\sqrt{5}}\right)$ by replacing all occurrences of $x$ by $0$
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