If `lim_(xtoa) {(f(x))/(g(x))}` exists, then which one of the following correct ?

Options

A. both `underset(xtoa)limf(x)` and `underset(xtoa)limg(x)` must exist
B. `underset(xtoa)limf(x)`need not exist `underset(xtoa)limg(x)` exist
C. neither `underset(xtoa)limf(x)` nor `underset(xtoa)limg(x)` may exist
D. `underset(xtoa)limf(x)` exist but `underset(xtoa)limg(x)` need not exist

Answer

C

Explanation

If `f(x)=sin((1)/(x))" and "g(x)=(1)/(x)," then both "underset(xto0)limf(x)" and "underset(xto0)limg(x)" do nto exist, but "underset(xto0)lim(f(x))/(g(x))=0" exists"`