Calculate $\lim\limits_{x\rightarrow 0^+}x\log(1+x^{-1})$ Can someone help me out with $$\lim\limits_{x\rightarrow 0^+}x\log(1+x^{-1})?$$ I tried Taylor's expansion to no avail.

$$\lim_{x\rightarrow 0}\ln\left(1+\frac{1}{x}\right)^{x}=\lim_{x\rightarrow 0}\frac{\ln\left(1+\frac{1}{x}\right)}{\frac{1}{x}}=\frac{\frac{-\frac{1}{x^{2}}}{1+\frac{1}{x}}}{-\frac{1}{x^{2}}}=\frac{x+1}{x}=1$$
