Question

If lim f(x)=lim f(x)=-1,but f(2)=1,then lim f(x)
x->2^- x->2+ x->2

Answer 1

Here the left side and the right side limits are the same. Therefore, lim x->2 f(x) = -1.

Remember the function may have a different value at f(x). But the limit, if it exists, is what the left and the right limits indicate.

Answer 2

im confused.

Answer 3

This function has a discontinuity at x = 2. If we were to look at its graph we will see there is a hole in the curve at x = 2. Approaching the point x = 2 on the curve from the left we see the y value approaching -1. Approaching the point x = 2 on the curve from the right we also see the y value approaching -1. Since both the left and the right approaches the same number (-1), we say the limit exists for this function as x approaches 2 and it is -1.. But the value of the function is defined as f(2) = 1. So on the graph you will see a hole on the curve at x = 2 but above the hole you will see a point at (2,1) which belongs to the function. So the function is defined at x = 2 and it is 1 but the function is NOT continuous at x = 2 because f(2) is not equal to the limit as x->2 f(x).