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

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

Question
Limits and continuity
asked 2020-10-27
Use the conjugate process \(\displaystyle\lim_{{{x}\to{0}}}\frac{{\sqrt{{{x}^{{2}}+{5}}}-\sqrt{{{5}-{x}^{{2}}}}}}{{x}}.\)

Answers (1)

2020-10-28
The expression simplifies:
\(\displaystyle\frac{{\sqrt{{5}}\cdot\sqrt{{{1}+\frac{{x}^{{2}}}{{5}}}}-\sqrt{{5}}\cdot\sqrt{{{1}-\frac{{x}^{{2}}}{{5}}}}}}{{x}}=\)
\(\displaystyle\sqrt{{5}}\frac{{\frac{{\left({1}+\frac{{x}^{{2}}}{{5}}\right)}^{{1}}}{{2}}-\frac{{\left({1}-\frac{{x}^{{2}}}{{5}}\right)}^{{1}}}{{2}}}}{{x}}\)
\(\displaystyle\Rightarrow\sqrt{{5}}\frac{{{\left({1}+\frac{{x}^{{2}}}{{10}}\right)}-{\left({1}-\frac{{x}^{{2}}}{{10}}\right)}}}{{x}}=\)
\(\displaystyle\sqrt{{5}}\frac{{{2}\frac{{x}^{{2}}}{{10}}}}{{x}}=\sqrt{{5}}\frac{{x}}{{5}}\)
0

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