Why do some functions have asymptotes?

1 Answer
Sep 2, 2015

Answer:

Some functions have asymptotes because the denominator equals zero for a particular value of #x# or because the denominator increases faster than the numerator as #x# increases.

Explanation:

Often, a function #f(x)# has a vertical asymptote because its divisor equals zero for some value of #x#.

For example, the function #y = 1/x# exists for every value of #x# except #x=0#.

The value of #x# can get extremely close to #0#, and the value of #y# will get either a very large positive value or a very large negative value.

So #x=0# is a vertical asymptote.

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Often a function has a horizontal asymptote because, as #x# increases, the denominator increases faster than the numerator.

We can see this in the function #y=1/x# above. The numerator has a constant value of #1#, but as #x# takes a very large positive or negative value, the value of #y# gets closer to zero.

So #y =0# is a horizontal asymptote.