Question

Construct a rational function that has a vertical asymptotic at x=1 and a removable discontinuity at x=-3. Explain how you determined your answer.

Answer 1

Let's take this step by step.
Vertical asymptotes occur whenever your limit tends towards infinity (or negative infinity) or either side. So the function is undefined at the pt. This often means you have in the denominator (x -a) where a is the vertical asymptote.
Removable discontinuity implies the limit exists at the pt, say b, but the function value f(b) does not exist. This occurs when you have (x - b) in the denominator but it also appears in the numerator and so cancels out, so that the limit at b can be evaluated.

So we have a situation that could look like this:



Can you think of an example now using this information? (ignore the graph, it's just a visual guide)