Course:MATH110/Archive/2010-2011/003/Notes/Limits at a point (easy cases)

How do we justify this habit of plugging-in for most limits? Here's the discussion (or the concept map if you prefer).

First, we have a nice formal definition of limits of functions at a point. (There's a note about that here), that's the stuff that uses $\epsilon$ and $\delta$ to do the job. It's very precise but it's pretty hard to do any computations with.

This leads to the next step. Using that precise definition, we can do a few things and one of them is talk about continuous functions. (The note about continuous functions can be found here). Those are very nice functions because they behave as one would expect in terms of their limits. The fact that for a function $f$ to be continuous at the point $x=a$ means that

$\displaystyle {\lim _{x\rightarrow a}f(x)=f(a)}$

actually tells us that continuous functions are the ones that do what seems the most obvious to do (such a nice property makes them super nice of course).

So now we have two things. First a tool: limits, which seems very useful and then we have a property: continuity, which seems very desirable because it seems discontinuous functions are so weird. Where do we go next? Well, we try to find as much continuous functions as possible! So mathematicians go out there in the jungle and hunt them down. They come back with some very good news (theorems) saying two things:

• A bunch of functions are actually continuous. That usually means mostly that all polynomials are continuous, but furthermore, that all Basic Functions are continuous as well (wherever it makes sense).
• Continuity is preserved under most obvious operations you can do with functions. In other words, if you have two continuous functions, you can can add them, subtract them, multiply them, divide them, put them to any exponent, compose them and more, again, as long as whatever you're doing makes sense (don't divide by zero, don't take negative square roots and so on).

It's important to notice I said mathematicians did that because we didn't prove it and you're not asked to know why that stuff is true. What you're asked though is to know what it means for that stuff to be true and how to use it (especially the part where you're allowed to play with functions).

Now that we have all that, we can actually start computing a lot of limits! Indeed, if you already know that a function is continuous at a point, then by definition it means that the limit of that function at this point is simply the value of the function there. (Try to make sure you understand that argument very well.) In other words, if you know that the function $f$ is continuous at the point $x=a$, then you know that:

$\displaystyle {\lim _{x\rightarrow a}f(x)=f(a)}$

Which means you can compute the limit by simply plugging in the value!

This doesn't end our discussions on limits, it simply explains why for a lot of them, we can simply plug in. It doesn't help us at all for all indeterminate cases for example.