Limit of a function without using L'Hôpital Rule Prove that 

$$\lim_{x\to 0} \frac{\ln(x+1)}{x} = 1$$

without using L'Hôpital Rule.
This is just the derivative of $y=lnx$ at $x=1$ by using the classic definition of the derivative. And so the answer is $1$. In this way, L'Hospital is avoided.
What about this: write our the Taylor series at 0 for the numerator, cancel out terms, take the limit.