Finding $\lim_{x \to 0}\frac{1}{\log{x+1}}-\frac{1}{\log{x+\sqrt{x^2+1}}}$ I have to find $$\lim_{x\to 0} \left( \frac{1}{\log({x+1})}-\frac{1}{\log({x+\sqrt{x^2+1}})}\right)$$
using only notable limits no derivatives
I'm completely stuck, I tried 4 times, using logarithms property and notable limits and every time I come to a more complex situation. When I was close to the solution I figured out that my solution was totally wrong due to the incorrect use of notable limits. Please help me (I'm sorry for the numerous questions regarding calculus of limits but today I'm gonna find plenty of limits because tomorrow I'm going to have a test..)
I managed to arrive to this form: lim (1/x-(2)/(x^2+2x)) using correctly the notable limits but I can't continue from here...
 A: I know this isn't a solution along the requested lines, but it can give some ideas.
Set $t=\log(x+\sqrt{x^2+1})$ so that $x=\sinh x$ and the limit becomes
$$
\lim_{t\to0}\left(\frac{1}{\log(1+\sinh t)}-\frac{1}{t}\right)=
\lim_{t\to0}\frac{t-\log(1+\sinh t)}{t\log(1+\sinh t)}
$$
Now $\log(1+u)=u-u^2/2+o(u^3)$ and $\sinh t=t+o(t^3)$ so
$$
t-\log(1+\sinh t)=t-\sinh t+\frac{1}{2}\sinh^2t+o(t^3)=
t-t+\frac{1}{2}t^2+o(t^3)
$$
and therefore the requested limit is $\dfrac{1}{2}$.
(Note: checking with a calculator I get that the expression, for $x=0.001$, evaluates to $\approx0.49975$.)
A: If you have a notable limit of the form
$$\lim_{x\to 0} \left( \frac{1}{\log({x+1})}-\frac{1}{x}\right)=L$$
then you can conclude that
$$\begin{align}
\lim_{x\to 0} \left( \frac{1}{\log({x+1})}-\frac{1}{\log({x+\sqrt{x^2+1}})}\right)&=\lim_{x\to 0} \left( \frac{1}{x}-\frac{1}{{x-1+\sqrt{x^2+1}}}\right)\\
&=\lim_{x\to 0} \left( \frac{1}{x}+\frac{x-1-\sqrt{x^2+1}}{2x}\right)\\
&=\lim_{x\to 0} \left( \frac{x+1-\sqrt{x^2+1}}{2x}\right)\\
&=\lim_{x\to 0} \left( \frac{1}{x+1+\sqrt{x^2+1}}\right)\\
&=\frac{1}{2}
\end{align}$$
Note, the actual value of the notable limit, $L$, drops out right away, but as it happens, it's also equal to $1/2$.
A: Here is my solution:
lim (1ln(x+1) - 1/ln(x+sqrt(x^2+1)) 
lim (1/(x)-1/(lnx+ln(1+sqrt(x^2+1)/x)))
lim (1/(x)-1/(lnx+ln(1+x/2+1/x)))
lim (1/(x)-1/(lnx+ln(2x+x^2+2)-ln(x)-ln2))
lim (1/(x)-1/(ln(1+x+(x^2)/2)))
lim (1/(x)-1/(2x+x^2))
lim (1/(x+2))=1/2 
Using only notable limits regarding natural logarithm, square root and exponential.
