eli5: what are limits in calculus???

In: Mathematics

Limits are a number that no matter how close you get to it, you can never reach

EXAMPLE: cut an item in half and remove the other half. Do this action over and over again.. no matter how many times you do this, you’ll never reach 0. You’ll get closer and closer but you’ll never be able to get to zero.

Sometimes you have an expression that’s undefined for some value of x. For instance…

(x^2 – 5x + 6) / (x-2)

…doesn’t work if x is 2. You get a division by zero which we can’t handle. Limits give us a way to analyze stuff like that – basically we say, “ok, 2 doesn’t work, but what if x were 2.1, or 2.001, or 2.000001”.

We can actually do one better than that, and figure out how the function would behave at (2 + 1/infinity), i.e. infinitely close to 2, and the value there is called the limit.

(For the thing above, the limit should be -1. You can sort of see that if you plug in x=2.001, or x=1.999, to see how it behaves close to 2.)

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.

A “limit” describes the behavior of a function as the input values approach some number.

The limit of 1/x as x approaches inifinity, for example, is 0. This is because the closer x gets to infinity, the closer y gets to zero.

Likewise, the limit of x as x approaches zero is infinity; this is because the closer x gets to zero (from the negative side of the axis), the closer y gets to infinity.

Give it a try if you don’t believe me.