logarithmic limit in R^n

Question

How does one prove that $$\lim_{|x|\to 0} \frac{ \log(1+|x|^2)}{|x|^2} = 1$$

when $x$ is a vector in $\mathbb{R}^n$, without using the multivariate Taylor expansion?

Answer 1

Note that $x$ appears in the expression only via $|x|$. If you wish to be very formalistic, just note that the given expression is a function composition: $$\frac{\log(1+|x|)}{|x|}=f(g(x)),\qquad f(t)=\frac{\log(1+t)}{t},\quad g(x)=|x|$$ and do the limit of $f(t)$ as $t\to0$ using single variable methods.