Limits

Intro

Limits are the underlying base to Calculus. We do not usually look at them as we work, but they are in virtually everything we do. They are a strange idea for math as well, because they are based on our perception rather than some calculation we do.

To take a limit of a function, we say “The limit of a function f(x) as x approaches some number, a, the function tends to some y”. I’ll write this as lim(x->a) f(x). This means that we can trace the line of the function, and the limit is where it appears to go as we push x really close to a.

The basic way we find a limit is by observing the function on both sides of a. So, suppose a is 1, solve f(x) for x = 0.75, then 0.9, then 0.99, and so on with numbers that get closer to 1 without solving for f(1) until you can observe a pattern of where it seems to go. Once you get an estimate for one side, do the same with the other. In this case, x = 2, then 1.25, then 1.1, then 1.01, and so on. If the limit exists at a, then they should come out to be the same or really, really, really close.

Sample Problem

Let’s take the function f(x) = x^2 and find the limit as x goes to 1. So we are looking for:

lim(x->1) x^2

So what does x^2 seem to equal to as x approaches 1. This would be simple if we just plugged it in: f(1) = (1)^2 = 1. Let’s use this as a hypothesis of sorts until we actually take the limit, then I’ll explain why we need limits.

Solution

From the left side of 1, we can use x = .9, .99, and .999 which equal:

f(.9) = .81
f(.99) = .9801
f(.999) = .998001

These numbers certainly look like they’re approaching 1, but we still need to observe the other side. Let’s use x = 1.1, 1.01, and 1.001

f(1.1) = 1.21
f(1.01) = 1.0201
f(1.001) = 1.002001

Likewise, these numbers also seem to approach 1, so the limit is proven to exist and is equal to 1.

lim(x->1) x^2 = 1

Now, take it back a step. Why do we have to do this whole tedious routine? In this case, we honestly don’t, it’s just easier to observe something you’re certain of at first. There are plenty of functions that are not defined at certain points though. Consider this:

lim(x->1) 3/(x-1)

What does this limit equal? We can’t just plug 1 in for x this time because we’d get 3/0; it’s a point where the function does not exist. We may not be able to plug 1 in for x, but we can observe where the function goes as x approaches it. That’s why we have limits.



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