LIMITS & CONTINUITY



GENERAL OVERVIEW


1) LIMITS must be understood first. They are separate concepts, however, CONTINUITY depends upon the understanding of LIMITS.

2) A limit looks like this: lim f(x)
                                    x→a

It reads: the limit of f(x) as x approaches a

f(x) is a functional expression (could be simple or complex) involving x, a is a fixed point on the x-axis or +∞ or -∞.  For the infinity types, this means the x-values move to the far right on the x-axis forever (+∞) or to the far left on the x-axis forever (-∞).

3) For a limit to exist, the expression for f(x) must get closer & closer to ONE &  ONLY  ONE  VALUE      
If not, then  there is no limit.

4) You must examine the x-values close to a on BOTH SIDES of a, since f(x) could get close to different values when approaching a from either side of a. They are called the right & left side limits. These must be the same for a limit to exist. A little plus sign in the exponent spot on a is used for the symbol for the limit from the right & a little negative sign for the limit from the left.

5) The TI-83 can be used to get a limit or to verify that there is no limit (will show). However, this procedure is lengthy & may not be practical for time limit testing. So, other much shorter methods will be covered.  To use your calculator, code in f(x) & go to 2nd Graph (table) & start inserting x-values close to the a-value. Look at the y-values [values for f(x)] & note if they are getting closer to one value. Make sure you use values for x on both sides of a.

6) Remember, when dealing with a limit, we are not concerned about x reaching a, just the values of f(x) as x gets closer & closer to a from both sides of a. We may or may not reach our limit, we simply don’t care about that. We can have a limit without reaching it.
(the distance to a wall cut in half forever, the limit is the wall, but you will never reach it).

7) Once limits are understood, one can analyze CONTINUITY. Basically, this means CONNECTED. So, a connected curve would be a continuous curve. That means NO BREAKS in the curve. Which means, NO HOLES or JUMPS or SHOOTING UP or DOWN forever when x-values approach the point a on the x-axis or increase to the right or left forever (infinity moves).

8) To be connected at a point a, we need a functional value f(a) to be defined on the function. Also, to eliminate any JUMPS, we need the f(x) values getting close to this functional value as we approach a from both sides. That is, we need the limit of f(x) as x approaches a to equal f(a).

9) So, we see that the concept of CONTINUITY at a point depends upon the limit existing at a & being equal to the y value or functional value at a, namely f(a). If a function is not continuous a point a, we say it is DISCONTINUOUS there or the function has a DISCONTINUITY at a.

10) To understand certain concepts clearly, studying cases where they fail to hold only strengthens our understanding. So, these will be covered.

11) As in most math concepts, doing many different types of problems, related to the concepts, would be the main path for total understanding.