Finding the limit when there is the mantissa function involved

Question

I can't seem to be able to use any known limits and/or variable change for this limit. I just need something to start. (log is natural logarithm and M(x) is the mantissa function.)

Answer 1

We have $$\frac{\log(e^x+1)}{x+1}<\frac{\log(e^x+1)}{x+M(x)}\le \frac{\log(e^x+1)}{x}\ .$$ The limit of the LHS and the limit of the RHS are both $1$, so by the pinching theorem (sandwich theorem, squeezing theorem) the limit of the middle term is also $1$.