Question

how can I solve Limit[1/x - 1/(E^x - 1), x -> 0] without l'hôspital?

Answer 1

\[\lim_{x \rightarrow 0} (1/x-1/(e^x-1))\]

Answer 2

Use \[x\rightarrow 0\quad,\quad e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\cdots\]

Answer 3

\[\lim_{x\rightarrow 0}\left(\frac{1}{x}-\frac{1}{e^x-1}\right)=\lim_{x\rightarrow 0}\frac{e^x-1-x}{x(e^x-1)}=\]\[=\lim_{x\rightarrow 0}\frac{x^2/2+x^3/6+\cdots}{x^2+x^3/2+x^4/6+\cdots}=\frac{1}{2}\]

Answer 4

it is such a nice thing that the book we use in college does not present us with such properties.

if only I had more time to process that.