What causes the graph of a rational function to have a vertical asymptote?

Answer 1

When your input variable causes your denominator to equal zero.

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A rational function of a variable, x is of the form f(x)/g(x), the ratio of two functions of x.

Suppose g(x) has a zero at x = x0. That is, g(x0) = 0. If f(x0) is not also equal to 0 then at x = x0 the rational function would involve division by 0. But division by 0 is not defined. Depending on whether the signs of f(x) and g(x) are the same or different, as x approaches x0 the ratio become increasingly large, or small. These "infinitely" large or small values are the asymptotes of the rational function at x = x0.

If f(x0) = 0, you may or may not have an asymptote - depending on the first derivatives of the two functions.