B.5) One-sided limits

The idea of a limit can be broken down into two parts to give one-sided limits, a notion used for piecewise defined functions and limits at the endpoints of the domain of a function.

If we restrict our attention to x's that are close to a and are less than a, we are looking at the limit of f[x] as x->a from below. If f[x] has a limiting value (say ) as x->a from below, we write
f[x] = .
This statement f[x] = means that for x < a, as x->a, the value of f[x] gets closer and closer to.

If we restrict our attention to x's that are close to a and are greater than a, we are looking at the limit of f[x] as x->a from above. If f[x] has a limiting value (say ) as x->a from above, we write
f[x] = .
This statement f[x] = means that for x > a, as x->a, the value of f[x] gets closer and closer to

Clearly, if the two-sided limit exists, say f[x] = L, then both one-sided limits must equal L.
Likewise, if both one-sided limits have the same value (in other words, = = L), then the two-sided limit must also equal L.

B.5.a)

B.5.b)

Take f[x] = { 2 x if x > 0 and -2 x + 2 if x <= 0. }
Make a plot of f[x]. Discuss whether f[x], f[x], and f[x] exist. Explain why or why not for each limit.

Answer:

Clear[f]
f[x_]:= If[x > 0, 2x, -2x+2]

Here is the plot of f[x]:

Plot[f[x],{x,-3,3}]

The graph clearly shows that if we approach x = 0 from below, the limiting value of f[x] should be 2. But if we approach x = 0 from above, the limiting value of f[x] should be 0. Since we can't decide which value should be the correct one, we say that f[x] does not exist. Since the one-sided limits both exist, we can write:
f[x] = 2;
f[x] = 0;
f[x] does not exist.
When this situation happens---the graph of f[x] approachs two distinct values as x-> a, we say the graph has a jump in it at x = a.