The discussion revolves around approximating the expression sqrt(x^2 - l^2) - l using the form x^2/(2l) under the condition that x is much smaller than l (x << l). Participants are exploring the validity of this approximation and the algebraic manipulation involved.
The discussion is active with participants clarifying assumptions and exploring different algebraic approaches. Some guidance has been offered regarding algebraic manipulation, but there is no explicit consensus on the correct interpretation or method yet.
There is a noted confusion regarding the expression under the square root, with participants questioning whether the original expression was stated correctly given the condition x << l.
If [itex]x<<\ell\,,\[/itex] then [itex]\sqrt{x^2 - \ell^2}\[/itex] is undefined.MahaX said:I need to show that: [itex]sqrt(x^2 - l^2) - l ≈ {x^2}/{2l}[/itex]
2. That should be valid for x << l
3. First I've tried to isolate l in the sqrt, but it got me nowhere. Anyone could show me a simple way to solve this?
SammyS said:If [itex]x<<\ell\,,\[/itex] then [itex]\sqrt{x^2 - \ell^2}\[/itex] is undefined.
Did you mean [itex]\sqrt{\ell^2-x^2}\ ?[/itex]
MahaX said:Sorry, I mean [itex]\sqrt{\ell^2+x^2}\[/itex]