Question

consider the following functions where a and b are unspecific constants f(x)=x^2+ax+b/x-1, is the line x=1 necessary a vertical asymptote, explain

Answer 1

For vertical asymptote of a function denominator of the function should be zero.

But here if (x^2+ax+b) has a factor of (x-1) i.e if (x^2+ax+b) has solution 1 than (x-1) in numerator and (x-1) in denominator gets canceled which ends up with (x-1) not contributing to the vertical asymptote. So (x=1) is not necessarily a vertical asymptote unless provided that (x^2+ax+b) does not have factor of (x-1).So x=1 is not necessarily be a vertical asymptote for f(x).