Can I have help in order to fully understand cerain limits of a function? ?

Question

First to be more familiar with the Ti-89 graphing calculator how do I know how to set the right limits for a table in a limits functions, for example: lim f(x)=?, w/ x approaching 0-?

I need to understand algebraically why the lim(limit) g(x)= DNE, with x approaching -2.

and why the function g(-2)=1?

lim g(x)= DNE, w/ x approaching 2?

g(2)= DNE?

g(0)= DNE?

lim g(x)= DNE, w/ x approaching 0?

lim g(t)= DNE, w/ t approaching 0?

lim g(t)= DNE, w/ approaching 2?

g(2)= 1?

please explain if you can in order for me to get the concept of limits...

Answer 1

first off, don't use your calculator to find limits. I know the 89 is great and all, but limits are too easy. Plus usually you'll have to prove your answer and I'm sure you're well aware a calculator is not a valid method.

secondly, I'm not exactly sure what your question is.

but I'll try...

A limit is the value f(x) gets CLOSE to from the LEFT and RIGHT as x goes to a number, c.

There are 3 cases.

1) you can just get the limit by plugging in c, where f(c) is the limit



   ex)    lim(x-->2)  2x - 3 = 2(2) - 3 = 1

  those are the stupid limits that no one cares about

Case 2) The limit does not exist. This happens when you plug in the number c and you get an undefined situation.

ex)   lim (x-->2)  1/(x-2) = 1/0     DNE

whenever you get a number divided by 0, the limit does not exist. wherever there's a vertical asymptote you have DNE

Other times you get DNE is when you have a piecewise defined function and there is a nonremovable discontinuity (a jump) where the limit from the left does not equal the limit from the right.

Case 3) The limit exists but it takes some work to show...

ex)  lim (x-->2)  (x^2-4)/(x-2)

if I plug in 2 I get 0/0.  WHENEVER I GET 0/0 THAT MEANS THE LIMIT EXISTS AND I HAVE TO DO SOME ALGEBRA TO FIGURE IT OUT. This is the case that's most common. Graphically there is a 'hole' at x = 2

there are other little exceptions with some trig graphs, but you should memorize those

hope this helps