How can you tell when a function has no vertical asymptotes?

Question

for example (2x+1)/(x^2+x+1) has no vertical asymptotes but why?

Answer 1

The rational function is f(x) = (2x + 1)/(x² + x + 1)

Vertical asymptote can be found by making denominator = 0.

x² + x + 1 = 0

Compare it to quadratic equation ax² + bx + c = 0.

a = 1, b = 1, c = 1

Discriminant b² - 4ac = (1)² - 4(1)(1)

= 1 - 4 = - 3

b² - 4ac < 0

The discriminant is negative then the equation x² + x + 1 = 0 has two imaginary solutions.

Real solutions are zeros of the function.

Since the denominator has no zeros then there are no vertical asymptotes.