Question

What is the asymptote of the graph of an exponential growth or exponential decay function? Explain your reasoning.

Answer 1

Let's consider an exponential decay function.

As an example, let's look at
`f(\text{x}) = 100(0.5)^{\text{x}}`

Here is a table of selected values:

`\begin{array}c \cline{1-2}\text{x} & \text{f(x)}\\\cline{1-2}0 & 100\\\cline{1-2}1 & 50\\\cline{1-2}2 & 25\\\cline{1-2}3 & 12.5\\\cline{1-2}4 & 6.25\\\cline{1-2}5 & 3.125\\\cline{1-2}6 & 1.5625\\\cline{1-2}\end{array}`

As x gets bigger, y = f(x) is slowly approaching y = 0. It will never get to this exact value because we keep taking half of the previous y value (eg: we go from y = 25 to y = 12.5); there's no way to arrive at zero through this continuous halving process. If x/2 = 0, then x itself must be 0. But we started off with a nonzero value.

Visually, the curve is slowly approaching the x-axis. We say that the horizontal asymptote is y = 0 which is directly overlapping the x-axis. Think of the asymptote as an electric fence you can get closer to but not actually touch.

See the graph below. I used Desmos to create the graph.