In this lesson, our instructor Vincent Selhorst-Jones teaches the idea of a limit. He discusses the fuzzy notion of a limit, exploring limits such as the ordinary function, piecewise function and a visual conception. He also defines a limit and explains how to find limits.
This lesson marks our entry into an entirely new section of mathematics: calculus. From here on, this course will preview some of the topics you will be exposed to in a calculus class.
We can conceive of a limit as the vertical location a function is "headed towards" as it gets closer and closer to some horizontal location. Equivalently, a limit is what the function output is "going to be" as we approach some input.
For notation, we use
to say, "the limit of f(x) as x approaches c."
Another way to conceive of a limit is to imagine "covering up" the location we're approaching with a thin strip. With that location covered, we ask, "Where does it look like this function is going (now that we can't see where it actually winds up)?"
Definition of a limit: If f(x) becomes arbitrarily close to some number L as x approaches some number c (but is not equal to c), then the limit of f(x) as x approaches c is L. Symbolically,
f(x) = L.
In the above definition, there are two important things to note:
We are looking at f(x) as x→ c, but we are not concerned with f(x) at x=c.
When we consider x approaching c (x→ c), we are considering x approaching from all directions, not just one side. To have a limit, f(x) must go to the same value from both sides.
It should be pointed out that the above is not technically the formal definition of a limit. This will certainly be enough for now, but if you're curious to know about it, check out the next lesson, Formal Definition of a Limit. [But you won't
need to understand that for a couple years (if ever), probably.]
Limits do not always exist. If the two sides don't settle down towards the same location, there will be no limit.
There are three main ways to find limits:
Graphs: Look at a graph of the function, and figure out if it makes sense to have a limit at the location. If so, find out what value the graph indicates. [This method is not very precise, but it gives a pretty good idea.]
Tables: Make a table of values where the input values get really close to the location we approach in the limit. If the output values settle down towards a single value the closer we get to the location, there is a limit. [This method is more precise
and can give very good approximations, but is still not perfectly accurate.]
Precise Methods: Later on, we will see algebraic methods to find precise values for limits. The lesson Finding Limits will go over this in detail. For right now, though, we'll just stick to the methods of graphing and making tables.
Idea of a Limit
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.