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# Functions, Graphs, and Limits

# Finding Vertical Asymptotes

Vertical asymptotes most frequently show up in rational functions. When a rational function *f*(*x*) has a non-zero constant in the numerator and an expression with a variable in the denominator, the function *f*(*x*) will have vertical asymptotes at all values of *x* that make the denominator 0. If the denominator has no roots, then *f*(*x*) will have no vertical asymptotes.

This works for the same reason that 1 / x has a vertical asymptote at zero: the numerator is a non-zero constant and the denominator is getting smaller and smaller, therefore the fraction will get bigger and bigger.

If we use a different constant, the principle is still the same.

If a rational function has something besides a non-zero constant in the numerator, we may need to be creative: factor the numerator and denominator, cancel if possible, and find the roots of whatever polynomial remains in the denominator after factoring.

Be Careful: Remember to simplify the rational functions!