 Limits form the heart and soul of calculus, so let's take a look at limits. So the formal definition of a limit is a nice, compact, mathematical statement. That's almost incomprehensible unless we've already taken calculus, as well as several post-calculus mathematics courses. On the other hand, the intuitive informal idea of a limit is a lot easier. The problem with intuitive informal ideas is that while we know what we're talking about, it's hard to describe. So here it goes. Informally, given some function f and some value a, the limit of f as x approaches a is the number l that f of x is close to when x is close to a. Now there's an important qualifier here. This number might not exist, so we have to add if such a number exists. We write this as l equals the limit x to a f of x, and we read this as the limit as x goes to a of f of x is l, or f of x converges to l as x goes to a, or the limiting value of f of x as x goes to a is l. And it is vitally important to remember that when we're talking about limits, the value of f of x at x equals a is not relevant to the limit as x approaches a of f of x. As with many things that are intuitive, it's actually harder to talk about limits than to actually find them. So for example, let's say I want to find the limit as x gets close to 1 of 2 to the power x minus 2 over x minus 1. And what we're looking for is we're looking for the limiting value of this expression as x gets close to 1. One way we might try to do this is we can try to find the limit numerically by evaluating the expression for x values close to 1. So let's try and find the value of this expression for x values that are close to 1. Oh heck, let's write on top of it at x equal to 1. Oh wait a minute, we can't do that. The value at x equals 1 is irrelevant to the value of the limit. So we want to pick a number that's close to 1 but not actually equal to 1. How about 0.9? So I'll let x equal 0.9 and evaluate our expression and we find the value of approximately 1.3393. Now if I was a politician or a shock jock political commentator, I would make a bold claim from one piece of evidence. I might even reject all evidence and say that the game is rigged because I can't let x equals 1. But if you're a normal person or a mathematician, you ask for a little bit more evidence. So let's pick another value that's close to 1. How about x equals 0.99 and evaluate our expression. And this time we get a value of 1.3815. And let's try one more value. How about x equals 0.999? And if we substitute this in, we get a value of 1.3858. And three pieces of evidence is good. However, they do share a common feature 0.9, 0.99 and 0.99 while they're all close to 1. They're all less than 1. And so the risk we run is that because all of these values share the same characteristic, namely that they're less than 1, it's possible that something may happen for them that doesn't happen for everything else. To understand this, we may have to look around. And in this particular case, we could be close to 1, but more than 1. So we should check out what happens in such cases. So I'll pick a value of x close to 1. How about 1.1 and I'll evaluate our expression. And we get approximately 1.4355. Again, we don't like generalizing from one example, so I'll pick another value close to 1. How about 1.01 and how about x equal 1.001? Now at this point, we've collected a whole bunch of data. And now comes the challenging part. From the data, we want to draw a reasonable conclusion. So here's where the problem is. If you have a lot of data, it's sometimes difficult to figure out how to make use of it. This is one of the reasons why politicians prefer only dealing with one fact at a time, or even better, they prefer using no facts at all to draw their conclusions. But as ordinary human beings and as mathematicians, we have the data. Let's see what we can do with it. And here are some guidelines that we'll use. We'll find at least six function values, four values that are close to our limit point. So here they are. Let's round all of these to the same number of decimal places. Now the number of decimal places we round to is going to vary. So that's where the artistry comes in. But if we round all of these values to the nearest whole number, it turns out they're all equal to 1. If all of these rounded values agree, then a limit probably exists. And at that point, we take a look at the two values that are closest to the limit point, and we round those two values, not the rounded values, but the original values, to some number of decimal places, and keep the maximum number of places that agree. So all of our rounded values agree. So there's probably a limit. If I round the values for 0.999 and 1.001 to one decimal place, I get 1.4. If I round them to two decimal places, I get 1.39. And if I round them to three decimal places, they disagree. So I'll leave them rounded to two decimal places and conclude that these values suggest a limit of around 1.39. Let's try another one. Let's find the limit as x gets close to 0 of sine of 1 over x. So we'll consider the values of sine of 1 over x when x is close to 0. So if x is 100, well, we don't actually care. 100 isn't actually close to 0. But if x is, say, 0.1, we find that sine of 1 over 0.1 is negative 0.5440. And again, if you want to go into a career in politics or political commentary, one piece of evidence is all you need, sometimes even less. But if you want to succeed in mathematics or in life, then you may want to take a look at more than one piece of evidence. So let's try x equals 0.01. And how about x equals 0.001? And again, all of the values we've chosen are close to 0, but they're all larger than 0. So we should look at x values that are close to 0, but less than 0 as well. So if x equals 0, the value of sine of 1 over x is not relevant. And that's a good thing because it's undefined in any case. How about x equals negative 0.1, or negative 0.01, or negative 0.001. And if we gather our data into a table of values for sine of 1 over x, we stare at the table, and no matter how we round these numbers, there's no way we can round these numbers so they will all agree on a single value. And because of that, we don't believe that sine over 1 over x approaches any single number as x gets close to 0, so we say that there is no limit. Now, there are some important limitations to this numerical method of finding limits. In some cases, we get something that might be a limit, and in other cases, we get a situation where we don't really seem to have any particular limit. So we might make the following general rule. If the numerical method suggests the limit doesn't exist, it's probably correct. On the other hand, if the numerical method suggests the limit does in fact exist, it means that there is more work ahead. We might not be able to do anything else with the problem, but we should always recognize that there is more that we need to do.