Question

How do i find vertical asymptotes ?

Answer 1

Vertical asymptotes occur when the function divides by zero and the limit is infinity. For example,
\[f(x) = \frac{x+1}{x-1}\]
has a vertical asymptote at x=1.
Be careful to distinguish between VAs and removable discontinuities:
\[f(x) = \frac{x+1}{x^2-1}\]
Setting the denominator equal to zero gets x^2-1 = 0 with solutions x = 1 and x = -1, but
\[ \lim_{x\rightarrow -1} \frac{x+1}{(x+1)(x-1)} = \lim_{x \rightarrow -1} \frac{1}{x-1} = -1/2\]
so one of those "divide by 0"s is just a hole, not a vertical asymptote. We can't cancel away the x-1 in the denominator, so as x goes to 1 the function blows up to infinity, giving us a VA.

Answer 2

Thanks