# Evaluate $\lim\limits_{y\to 0}\log(1+y)/y$ without LHR or Taylor series

Problem 12 on p. 297 of Spivak's Calculus (first edition) is

Find $\lim\limits_{y\to 0}\log (1+y)/y$. (You can use L'Hospital's rule, but that would be silly.)

I'm not sure the other method he's looking for besides LHR. I can also think of expanding $\log(1+y)$ in Taylor series, but this is before the section of the book on Taylor series.

$$\lim_{y\to 0} {\log(1 + y)\over y}= \lim_{y\to 0} {\log(1 + y) - \log(1)\over y} = \log'(1) = 1.$$
$$\lim_{y\to 0} \frac{\log(1+y)}{y} = \lim_{y\to 0} \log(1+y)^{1/y} = \log \lim_{y\to 0} (1+y)^{1/y} = \log e = 1$$
• @user127.0.0.1 - he is using the rule $a \log(b)$ = $\log(b^a)$ with $a = 1/y$ and $b = 1 + y$. – breeden Jan 25 '14 at 1:49