# Give an example of a function with both a removable and a non-removable discontinuity.?

Relevance

Let's consider the function f(x) = (x + 1)(x + 2) / (x + 2)

Now, since there's and x+2 in the denominator, we can't evaluate the function at x = -2. That would make the denominator zero, and we can't divide by zero. There's a discontinuity there.

But we can cancel out x + 2 in the numerator and the denominator. So this is a "removable" discontinuity - the function is indistinguishable from g(x) = x+1, except for the discontinuity. Note that, as you get close to x = -2, f(x) doesn't go to infinity. It just approaches the "hole" in the function.

A non-removable discontinuity is one that you can't get rid of by canceling. So, h(x) = (x+1) / (x+2) has a non-removable discontinuity at x = -2. As you get close to x = -2, the value of h(x) blows up.

I hope that helps!

Edit: It occurs to me that maybe you mean a single function that has both a removable and a non-removable discontinuity. f(x) = (x + 2) / [(x + 2)(x + 1)] would work for that - the discontinuity at x = -2 is removable, and the discontinuity at x = -1 is non-removable.