Question

Why do some functions have asymptotes?

Answer 1

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.