could be anything, we don’t know. does not exist as it is undefined.
Remember the open dot on the -axis means the function fails to exist for .
Visual introduction to Limits
Video Lecture
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Explanation
This process of considering values near a point, but not actually at a point was the original motivation behind limits. Before we get into the nuts and bolts of what limits are and how to compute them though, it is helpful to get a better idea of what a limit actually is. That is, what does it look like when we “take a limit”?
As we progress through the semester (and future calculus semesters) it is deceptively easy to lose sight of what is actually happening when we are computing a limit. Indeed, with all the number crunching and simplifying of functions and expressions, it can easily feel just like another algebra class (albeit with very difficult algebra!) However, this misses the very essence of limits! The key idea about limits isn’t what is happening at a point (say, in time), but rather what is happening around a point (in time).
Consider the following situation.
Imagine: you are home alone with your pet cat, enjoying the billiards trickshot championships on TV. You are watching one of the players lining up a shot which will hit (and hopefully sink) three balls simultaneously. Right as the player’s cue hits the cue ball toward the meticulously placed trio of billiard balls however, your cat decides to swipe out a paw and knock your bottle of soda off the coffee table, splashing its contents all over your leg, shoe, sock, and floor.
Naturally you jump up with a yelp of both surprise and frustration.
Your cat jumps off the coffee table with a dismissive flick of its tail (how rude of you to yell when it was just getting comfy on the now spacious table!) and just as you are considering what size coffin your feline irritant might need you hear the CRACK of the cue ball hitting the setup. Too late, you glance back to the tv just as the billiard balls all perfectly drop into their assigned pockets.
You missed the shot, and now you have a mess to clean up. The cat has also managed a strategic retreat, likely not to be found until you have conveniently cooled down.
So, what does the above have to do with math, let alone limits you ask? Well, let’s consider for a moment what it is you missed on
the TV. On the one hand, you definitely didn’t see the shot, but on the other hand you could probably figure out how it worked
right? After all, objects in the real world don’t teleport around, so the cue ball had to traverse the billiard table from where it was
sitting while the player lined up the shot, to where the three billiard balls were setup at the start of the shot.
Moreover, you know the outcome - the three balls each went to the three pockets, so you know the cue ball had to
hit the balls in exactly the right spot in order for that to happen. Thus, without actually seeing the event, you
can mentally recreate what must have happened, given the few specific moments you saw both beforehand and
afterwards.
Believe it or not, this reconstruction of a moment you missed, by extrapolating from information both before and after the moment of interest, is exactly the idea of a limit!
It is important to realize however, that had you missed more of the beginning and/or more of the ending to that particular shot, it could have been impossible to know what happened. Had the cat knocked the drink over before the shot was lined up, or if you hadn’t looked back in time to see the balls sinking into the pockets, you couldn’t know how the cue ball hit the billiard balls, or where they went. So you needed to see what happened close to the moment (both before and after) when the cue balls hit the billiard balls, but you didn’t need to actually see what happened at the moment. This is both the strength, and weakness, of limits.
At its core taking a limit of a function at some specific value , is the process of figuring out what should be, by finding out what is for values “suitably close” (remember this phrase, it will come up again! And again... and again...) to .
As a more concrete visual, consider the following graph:
1 : What is the value of ?
The previous question is, admittedly, a little goofy; a better question might be the following:
2 : What value is the function approaching as gets “near” to .
could be anything, we don’t know. does not exist as it is
undefined.
Even though the function doesn’t exist at , it does get infinitely close to the value as gets close to .
There are many times when a function’s behavior at a point does not necessarily correlate to its behavior right nearby (often because it doesn’t exist!). This is why we use limits! Remember, limits tell you about a function’s behavior arbitrarily close to the (-)value of interest, but not actually at the value. This let’s us circumvent the problems that arise when a function doesn’t exist at some specific -value.
3 : What is the purpose of a limit?
To give me a headache. To find what a function equals at a given point. To determine where a function is heading for a specific
-value of interest (without knowing what it equals there). To find derivatives and integrals and all kinds of calculus-y things.
Our goal is to make this process rigorous, i.e. to be able to write down in the language of mathematics this idea of limits and how to compute them. Before we can do this however we need to get a little more specific about what we mean by having an value “nearby” to the value of interest. We do that in the next part.