eli5: what are limits in calculus???
In: Mathematics
Regarding one-sided limits versus two-sided limits: Say you’re curious about how the output of some function behaves as x gets close to zero. You might start with something small and positive like x=0.1. You check out what f(0.1) looks like. Then you tick down: f(0.01), f(0.001), f(0.0001), etc, observing the pattern if there is one at all. This is the right-sided limit of f as x approaches zero, because you’re walking towards zero “from the right” on a standard number line. This answers the question, “what does the output of f do as the inputs DEscend to zero?”
On the other hand, you could instead start with x=-0.1 and walk up towards zero. This would be the left-sided limit, because you are approaching zero from the left on a standard number line. This answers the question, “what does the output of f do as the inputs Ascend to zero?”
Suppose you do both of the above and get two answers. If they’re the same answer, then one might say the two one-sided limits “agree”. (A classic example: f(x) = sin(x)/x. Both one-sided limits are 1 as x approaches zero, though this is tough to prove.) In this case, the two-sided limit is defined to be equal to that common answer. This answers the more general question, “what does the output of f do as x gets close to zero?” If, in contrast, the two one-sided limits do NOT agree, then the two-sided limit is said to be undefined. (Classic example: f(x) = 1/x.) It has no answer. The anecdote I share with students is about when my sister and I would be in the back seat of the van on road trips. We had one of those tiny travel TV’s with a built-in VHS player. If we could agree on a movie to watch, Mom would put it in. If we couldn’t agree, we would watch nothing. So it is with limits: if the two one-sided limits agree, then the two-sided limit has that answer as well. If the two one-sided limits don’t agree, then the two-sided limit doesn’t have an answer at all; it is undefined.
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