Our topic of discussion in this section is **one-sided limits**,
which builds upon the preceding lesson on continuity. Create a new worksheet
called 03-One-Sided Limits. The basis of one-sided limits is that when a function
jumps suddenly from one value to another, it often is not possible to describe
the function's behavior with a single limit. What we can do, though, is to
describe the function's behavior *from the right* and *from the left*
using two limits. Consider the following graph, the code of which is provided:

p1 = plot(-x^2+6, x, 0, 2) p2 = plot(x-1, x, 2, 4) pt1 = point((0, 6), rgbcolor='black', pointsize=30) pt2 = point((2, 2), rgbcolor='white', pointsize=30, faceted=True) pt3 = point((2, 1), rgbcolor='black', pointsize=30) pt4 = point((4, 3), rgbcolor='black', pointsize=30) (p1+p2+pt1+pt2+pt3+pt4).show(xmin=0, xmax=4, ymin=0, ymax=6)Toggle Explanation Toggle Line Numbers

1-2) Plot -x^{2}+6 from 0 to 2, x-1 from 2 to 4

3-6) Create three closed points, one open

7) Combine the plots and points, then show the result with the given x and y boundaries

The above function has a discontinuity at x=2, and since the two pieces of the function approach different values:

is undefined.

You probably see where this is going. What we *can* say that the limit of
f(x) as x approaches 2 from the left is 2, and the limit of f(x) as x approaches
2 from the right is 1. If you were to write this, it would look like:

and

The minus sign indicates "from the left", and the plus sign indicates "from the
right". Since the limit of f(x) as x approaches 2 from the right is equal to f(2),
f(x) is said to be *continuous from the right at 2*. The limit of f(x)
as x approaches 2 from the left does not equal f(2), however, so f(x) is not
continuous from the left at 2.

One-sided limits are usually fairly straightforward. However, be aware
that when a function approaches a **vertical asymptote**, such as
at x=0 in the following graph, you would describe the limit of the function as
approaching -oo or oo, depending on the case. A vertical asymptote is an x-value
of a function at which one or both sides approach infinity or negative infinity.

plot(1/x, x, -6, 6, randomize=False).show(ymin=-5, ymax=5, xmin=-5, xmax=5)Toggle Explanation Toggle Line Numbers

1) Plot 1/x from -6 to 6. randomize=False produces a more consistent result when this particular function is plotted.

Here, we would say that the limit of f(x) as x approaches zero from the left is negative infinity and that the limit of f(x) as x approaches zero from the right is infinity. The limit of f(x) as x approaches zero is undefined, since both sides approach different values. Visually,

,
, and
is undefined.

Refer to the following graph of f.

(a) At which points is f discontinuous?

(b) For each of these points, determine whether f(x) is continuous from the right, from the left, or neither.

(c) Classify any points of discontinuity as either removable or nonremovable.

(a) Toggle answer

(b) Toggle answer

(c) Toggle answer

(a) At which points is f discontinuous?

(b) For each of these points, determine whether f(x) is continuous from the right, from the left, or neither.

(c) Classify any points of discontinuity as either removable or nonremovable.

(a) Toggle answer

(b) Toggle answer

(c) Toggle answer