In this lesson, our instructor Vincent Selhorst-Jones teaches you how to find limits. Youll learn the method for normal functions, covering canceling factors, rationalization, and piecewise functions. The lesson ends with six examples to give you plenty of guided practice.
The easiest limits to find are the limits of "normal" functions. Now "normal" is not a technical term, it's just supposed to mean the kind of functions we're used to dealing with. Functions where
It does not "break" at the point we're interested in (the function is defined and makes sense);
It is not piecewise and/or the point we're interested in is not at the very edge of the domain.
Assuming the two conditions above are true for a function and the point we're interested in is the value that x approaches in the limit, it will almost always be the case that the limit for the function is the same as the value for the function.
The above is true because, in general, most of the functions we're used to working with don't do anything "weird". That is, they're defined everywhere, they don't have holes, and they don't jump around. Therefore the functions we're used to working with
go where we expect them to go.
Iff(x)is "normal" aroundx=c,then
f(x) = f(c).
This is true even if f(x) has "weird stuff" happening somewhere else. All we care about is x→ c, so as long as the neighborhood around x=c is normal, this works.
Often we can't use the above because something "weird" does happen at x=c. A common weird thing is dividing by 0. In this case, we can sometimes find the value of the limit (if it exists) by canceling factors before taking the limit.
For a radical expression (one with a root), the conjugate is the same expression, but flipping the sign on one side:
x2 − 3x
x2 − 3x
Conjugates allow us to rationalize fractions, which can sometimes help us find the value of a limit. By multiplying the top and bottom of a fraction by the appropriate conjugate, we can often find limits that would not otherwise be possible.
We'll discuss evaluating the limits of piecewise functions in the next lesson, Continuity and One-Sided Limits. For now though, remember: as long as you're not trying to evaluate a limit on a piecewise "breakover" (where it switches from one
function piece to another), the function is probably behaving "normally" on the pieces that contain the point you care about. Thus, if it's not a breakover point, you can approach it like you're evaluating the limit of "normal" function: just plug
in the appropriate value and see what comes out.
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.