Calculus #17: Early Transcendentals 2.2B

In the last section, we noted how the limit can be different depending on whether you approach the function from the left or right. Let’s continue.

One-Sided Limits

Before I get into this, I want to once again take pains to point out that “left-right” is just a way to visualize what’s happening. For a limit as x goes to 0, another way to say “x approaches 0 from the left” would be “as x increases toward 0.” Another way to say “x approaches 0 from the right” would be “as x decreases toward 0.”

Here’s a real world example to help you remember. Imagine I just tied you to a tree and lit a fire under you. The level of pain (in metric AAAHs) you experience will rise as a function of time, f(t) = t^2 as the fire grows and becomes hotter. Say we also know that you’ll die of shock at 100 seconds in. Now, we want to know how much pain you’re in at 100 seconds.

Well, after 100 seconds, you’re in no pain, because you’re dead. So, if we measure going backward in time from, say, 110 seconds (that is, we go from right to left), we’ll predict that you’ll feel no pain at exactly 100 seconds. If we go from the left to the right, we expect you’ll experience 10 kiloAAAHs at 100 seconds.

Quite a difference. Fortunately, the math here is fairly straightforward. For now, you need to understand 2 things: (1) How to denote the direction of the direction (“handedness”) of a limit. (2) A simple rule for knowing when handedness matters.

Here’s how you denote a limit coming from the left hand side of the graph (i.e. increasing toward the limit):

\lim_{x \to a^-} f(x) = L

The only difference is we put a little minus sign next to the a. The easy way to remember this is that the left quadrants of a traditional Cartesian graph or the negative quadrants. We’re coming from the negative side, so to speak.

You can probably guess the right hand side limit.

\lim_{x \to a^+}f(x) = L

Fortunately, any time you have a smooth curve (i.e. no discontinuities or jagged points), you don’t need to know whether you’re increasing or decreasing toward the dependent variable. In that case, you’ll get the same value either way, so you’re back to our earlier definition of a limit. Or, as the book says:

\lim_{x \to a} f(x) = L if and only if \lim_{x \to a^-}f(x) = L and \lim_{x \to a^+}f(x) = L.

More simply: If the left approach and right approach both approach the same value (in this case, L), you don’t need the little plus or minus sign.

Note: This is true even if they both approach a discontinuity. I’m harping on this because people get confused sometimes. Remember, the limit is what the the dependent variable approaches as the independent variable approaches some value. Or, as you’ll see it more often, the limit is what y approaches as x approaches some value.

Next subsection: Infinite Limits!

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