Calculus question?

Question

Given: f(x) = tan(x) / x

Evaluate the limit as x approaches 0. Is there any way to remove the discontinuity in f(x)?

I have to describe what happens to f(x) when x = 0 and it looks like there's just a hole in f(x). However, if I understand this right, holes are caused by removable discontinuities, while non-removable discontinuities cause vertical asymptotes. There is definately not a vertical asymptote on the graph of f(x) when x = 0, so I was hoping to remove the discontinuity and prove that there is just a hole.

Thanks.

Answer 1

tan(x) = sin(x)/cos(x)

so, tan(x)/x = sin(x)/cos(x)x = (sin(x)/x)/cos(x)

and hence,

the limit as x approaches zero would be

lim x->0 [sin(x)/x] * lim x->0 [1/cos(x)] = 1 * 1 = 1