Friday, January 8, 2016

real analysis - Find limxto0fracln(x2+1)x2 without L'hopital's rule



I have to find the limit without L'hopital's rule:
lim



Is it possible?
I thought about using squeeze theorem or something, but it didn't work out.




Hints are more than welcome!



P.S - I didn't study Taylor series or Integrals yet.


Answer



\begin{align} \lim_{x \to 0} \frac{\ln (x^2+1)} {x^2}&=\lim_{x \to 0} \ln (x^2+1)^{\frac{1}{x^2}}\\ &=\ln\left(\lim_{x \to 0} (x^2+1)^{\frac{1}{x^2}}\right)\\ &=\ln e=1 \end{align}



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