Question

Hi how do i solve this limit:
http://img641.imageshack.us/img641/9793/limit.gif

thank you

Answer 1

Rewrite the sqrt as a 1/2 coefficient, and bring the 1/2 outside the limit. Then rewrite as\[\frac{1}{2}\lim \frac{\ln[(x+1)/(x-1)]}{x}\]then apply L'Hopital's rule by taking derivative wrt x in both numerator and denominator to get\[=\frac{1}{2}\lim \left( \frac{-2}{x^2-1} \right)\]factor out the constant\[=-1*\lim \frac{1}{x^2-1}\]the limit of a quotient is the quotient of the limit\[=-\frac{1}{\lim(x^2-1)}\]Now the denominator becomes\[\lim_{x \rightarrow 0}\left( x^2-1 \right)=-1\]and so substituting -1 in the denominator above gives\[=-\frac{1}{\lim (x^2-1)}=-\frac{1}{-1}=1\]