Question

show using limits that f(x)=tan(x) is continuous at x=0

Answer 1

for any f(x) to be continuous at x=a,

lim x->a- f(x) = f(x) = lim x->a+ f(x)

what this means is, the function value at x=a must be approached from both the left and right.

Answer 2

So in your case of f(x) = tan(x) at x = 0,

Now using what we discussed earlier,

lim x->0- tan(x) = 0 since there is no problem plugging it straight in
tan(0) = 0
lim x->0+ tan(x) = 0 again, plugging it straight in.

Now since all three parts are equal, the function is therefore continuous at x = 0.