How would I go about answering this limit question?

Question

Give examples of two functions, the first has a lim f(x) = 8

x->2

and a second function for which lim f(x) does not exist.

x->2

Show all steps when necessary.

Answer 1

Hey Anne!

There are tons of different ways that you could approach this question, but here's how I'd probably go about it.

In the first case, we want the limit as x approaches 2 to exist, and we want the output at that point to be 8. There's more than one way for this limit to exist, but the easiest way would just be to choose a continuous function. Now, there's infinitely many different continuous functions you could choose, but why not just choose a straight line? Try to think of any straight line that passes through the point (2,8)

For the second, now we want to make sure the limit as x approaches 2 doesn't exist. If we wanted to show off, we could just make a piecewise function that doesn't match up at x = 2, but maybe a bit more simply, we could choose a function with a vertical asymptote at x = 2. So maybe consider the reciprocal function f(x) = 1/x, and how you might shift that function around so its asymptote might be at x = 2.

Hope that helps you get started!