Limit Notation in Calculus

This is going to be a very short post about a crucial subject in calculus: limts!  In my previous post, I gave a short and quite basic explanation of what is meant when we talk about limits in calculus.  I explained that they are similar to a concept that you know in everyday life, such as a speed limit, though they have their differences.  I also simply demonstrated that the value at a point on a curve does not necessarily equal the limit as you approach the point.  They require a different way to think about curves and graphs, in that where you are traditionally used to thinking about what happens exactly at a specific point on a graph (such as, what is the value of y when x equals 2), limits require you to consider what is happening to the curve as you approach a point and get infinitely closer to it.  Essentially, it may be thought of as "based on the shape of the curve that I know for sure, what does it look like the curve is going to do next?"  What it looks like it's going to do, the limit, might not always be exactly what it does do, which would be what the equation of the line would describe exactly.

So, that is a very brief overview of what my last post discussed, and over the next several posts, I will be expanding on this concept of limits, and explaining several different points about them and how to work with them.  First, however, I need to explain the notation for limits.  Limit notation in calculus is quite intuitive when you look at it.  That is, it is quite easy to understand when you see it.  Here is how you would write the symbol for a limit:

To read this, one would read it as "the limit of f of x, as x approaches a, equals L."

The "lim" part indicates that this is a limit, the "f(x)" (in this case) could be any function or expression, and the x→a part indicates the value "a" that x is approaching for which we want to evaluate the limit, L.  Let's consider the following graph again, the same one I drew in my introduction to limits post.

Now, say that we want to determine what the limit of this curve is (let's just call it the generic f(x)) as x→p (*note* assume for now that the point P is designated by the ordered pair (p,y)... tonight I'm too lazy to go back and edit my picture!).  We would write this expression as:

This expression reads as "the limit of f of x, as x approaches p."  Now, in this case, just by looking at the graph, we can visually see that if we trace the curve and move towards point P, where x=p, we can see that the curve is approaching y=4.  In other words, the limit is 4.

Similarly, refer back to my calculus limits introduction post, and you can see on the second example, where y=10 when x=p, the limit as x approaches p is also 4.

I hope that this series of posts is beginning to help you to grasp the concept of limits in calculus.  In my next post, I would like to develop this one a bit further, by introducing the concept of directional limits.  For the examples I've shown so far, we have only considered functions where the curve approaches the same point from both sides.  However, some graphs (e.g. piecewise equations) can have a different limit value, depending on which direction you approach "a" from.  I will explain these one-sided limits and their limit notation in the next post.

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