# Use the conjugate process lim_(x->0) (sqrt(x^2+5)-sqrt(5-x^2))/x.

Question
Limits and continuity
Use the conjugate process $$\displaystyle\lim_{{{x}\to{0}}}\frac{{\sqrt{{{x}^{{2}}+{5}}}-\sqrt{{{5}-{x}^{{2}}}}}}{{x}}.$$

2020-10-28
The expression simplifies:
$$\displaystyle\frac{{\sqrt{{5}}\cdot\sqrt{{{1}+\frac{{x}^{{2}}}{{5}}}}-\sqrt{{5}}\cdot\sqrt{{{1}-\frac{{x}^{{2}}}{{5}}}}}}{{x}}=$$
$$\displaystyle\sqrt{{5}}\frac{{\frac{{\left({1}+\frac{{x}^{{2}}}{{5}}\right)}^{{1}}}{{2}}-\frac{{\left({1}-\frac{{x}^{{2}}}{{5}}\right)}^{{1}}}{{2}}}}{{x}}$$
$$\displaystyle\Rightarrow\sqrt{{5}}\frac{{{\left({1}+\frac{{x}^{{2}}}{{10}}\right)}-{\left({1}-\frac{{x}^{{2}}}{{10}}\right)}}}{{x}}=$$
$$\displaystyle\sqrt{{5}}\frac{{{2}\frac{{x}^{{2}}}{{10}}}}{{x}}=\sqrt{{5}}\frac{{x}}{{5}}$$

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