Jorden Pace

2022-07-07

How to solve following limit
I've been struggeling a bit with the following limit:
$\underset{x\to 0}{lim}\frac{a-\sqrt{{a}^{2}-{x}^{2}}}{{x}^{2}}$
The solution is:
If a < 0 then $\mathrm{\infty }$. If a > 0 then $\frac{1}{2a}$
But I don't know how to get there.
Thank you

potamanixv

Expert

$\begin{array}{rl}\underset{x\to 0}{lim}\frac{a-\sqrt{{a}^{2}-{x}^{2}}}{{x}^{2}}& =\underset{x\to 0}{lim}\frac{a-\sqrt{{a}^{2}-{x}^{2}}}{{x}^{2}}\cdot \frac{a+\sqrt{{a}^{2}-{x}^{2}}}{a+\sqrt{{a}^{2}-{x}^{2}}}\\ & =\underset{x\to 0}{lim}\frac{{x}^{2}}{{x}^{2}\left(a+\sqrt{{a}^{2}-{x}^{2}}\right)}\\ & =\underset{x\to 0}{lim}\frac{1}{a+\sqrt{{a}^{2}-{x}^{2}}}\\ & =\frac{1}{a+\sqrt{{a}^{2}}}\end{array}$
Now the tricky part is how you simplify $\sqrt{{a}^{2}}$ base on the sign of a.

tripes3h

Expert

We have that
$\underset{x\to 0}{lim}\frac{a-\sqrt{{a}^{2}-{x}^{2}}}{{x}^{2}}=\underset{x\to 0}{lim}\frac{{a}^{2}-{a}^{2}+{x}^{2}}{{x}^{2}\left(a+\sqrt{{a}^{2}-{x}^{2}}\right)}=\frac{1}{a+|a|}$
from here you can do

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