I really need to understand wha tmakes a graph have these discontinuities
1)Removable
2)Non-Removable
and how to tell them in a function
Usually these occur with rational functions. A discontinuity occurs when the function is not defined. Usually that is when the denominator is zero.
f(x) = (x+3)/(x-3)
since division by zero is not defined, f(x) is not defined at x=3; there is a discontinuity there.
If the limit from both sides at the discontinuity exists is the same, then there is just a "hole" at the discontinuity. It may be removed by defining f(x) there to be equal to the limit.
f(x) = (x^2-9)/(x+3)
at x = -3, we have 0/0, which is not defined. However, everywhere else, f(x) = x-3, since
f(x) = (x-3)(x+3)/(x+3) = x-3
So, if we define f(x) = -6 at x = -3, then the limit on both sides is -6 and f(-3) is also -6, so f(x) is continuous there.
So, find discontinuities wherever f(x) is not defined.
Then, check the limit from both sides there. if the limits exist and are the same, the discontinuity is removable, unless f(x) is defined to have some other value there.