1.2 One-Sided Limits
The easiest way to approach limits is geometrically, via the simpler concept of one-sided limits. Before we get to a rigorous definition, it helps to first grasp the concept intuitively (i.e., visually!). The notation will be introduced in the first examples.
In the following examples, we’ll be using piecewise-defined functions (if you’re not familiar with this type of function, refer to my lecture notes on this topic before continuing). All the other types of function with which you're familiar with are too tame for our present purposes.
For examples 1 and 2, we’ll use the piecewise-defined function
with graph
Example 1: Find
The first of these strange looking expressions that we are being asked to “find” is called a right limit. To read the expression
, say
“the limit as x approaches 0 from the right, of f(x)” or
“the right limit of f(x) as x approaches 0”.Think of a bug crawling on the x-axis, approaching 0 from the positive side. Imagine the graph of f is above the bug’s head. Where is the graph (over the bug’s head) going just as the bug arrives at the origin? The answer is
Look at the graph above. As the bug approaches 0, the graph of f approaches the bug’s head, so the bug better not ever actually arrive at 0 or it’ll bump its head! This means that
,
for the height of the function’s graph as x approaches 0, is itself 0.
The only difference between the right limit just discussed, and the left limit
, is that with the left limit, the bug approaches 0 from the left, from the negative side of 0 on the x-axis. In this case, the graph of f will also bump the bug on its head, so we must also have that
Example 2: Find
Now our x-axis insect is approaching 1 on the x-axis, from the right and left. As the bug approaches 1 from the the right, what height is the graph approaching?
As you can see, the graph approaches a height of 2 as the bug approaches 1 from the right. In other words,
In words, “the limit of f (x) as x approaches 1 from the right”.
Now as the bug approaches 1 from the left, the graph approaches a height of 1:
As you can see, the graph approaches a height of 1 above the bug’s head as the bug approaches 1 along the x-axis from the left. Using our more impressive notation, we write:
The limit of f(x) as x approaches 1 from the left, equals 1.
In this example, notice that the right and left limits do not agree.
Also note that the limit of a function doesn’t have to be equal to the actual value (if any) of the function at the limit point. In this last example, the function was defined with f(1)=2. The function could have been defined to equal any other value at x=1, and the left and right limits would remain the same.
IGNORE the actual value of the function at the limit point when finding right and left limits. The question to ask yourself, as the bug crawling along the x-axis, is “Are you feeling lucky?”. Just kidding… actually you should ask “What is the natural continuation of the function as the bug approaches its destination?”.
Example 3: Find
This is just example 2 revisited. This time we’ll solve the limit algebraically rather than geometrically. We have
since f(x) = 3 – x for any x on the right side of 1 (that is to say, for x >1).
As x approaches 1, clearly (3– x) will approach 3–1=2, so we get
This agrees with our geometric intuition. As to the left limit, we have
since f(x) = x2 for any x on the left side of 1 (that is to say, for x <1). And so
1.2 Exercises Do all of them! :-)
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