Limit/sqrt(x+4)-2/x/Example

We consider the limit

$\operatorname {lim} _{x\rightarrow 0}\,{\frac {{\sqrt {x+4}}-2}{x}},$

where ${}x\in \mathbb {R} \setminus \{0\},\,x\geq -4$. For ${}x=0$, this term is not defined, and from this term one can not read of directly whether the limit exists. It is however possible to multiply the numerator and the denominator by ${}{\sqrt {x+4}}+2$, then we get

${}{\begin{aligned}{\frac {{\sqrt {x+4}}-2}{x}}&={\frac {{\left({\sqrt {x+4}}-2\right)}{\left({\sqrt {x+4}}+2\right)}}{x{\left({\sqrt {x+4}}+2\right)}}}\\&={\frac {x+4-4}{x{\left({\sqrt {x+4}}+2\right)}}}\\&={\frac {x}{x{\left({\sqrt {x+4}}+2\right)}}}\\&={\frac {1}{{\sqrt {x+4}}+2}}.\end{aligned}}$

Due to the rules for limits, we can determine the limit in the numerator and in the denominator separately, where for the denominator we use the continuity of the square root according to exercise. Hence, the limit is ${}1/4$.