also how is it that $\displaystyle csc0=\infty$ yet $\displaystyle csc0=\frac{1}{sin0}$ which equals 1/0 which is undefined, I think these problems are of the same vein.
i've also noticed on some calculators the sin of 0 gives me a really small number instead of 0.
"explanation" leaves a bit to be desired ...
$\displaystyle \lim_{x \to 0^+} \frac{1}{x} + \frac{1}{\cos{x}-1}$
saying it w/o using the oft misused word "infinity" ...
as $\displaystyle x \to 0^+$ , the first term becomes a very large, unbounded positive value
as $\displaystyle x \to 0^+$ , the second term becomes a very large, unbounded negative value
(large, unbounded positive value) + (large, unbounded negative value) is an indeterminate value ... hence the need to set up the limit ao L'Hopital can be used.