If they are equal and this theorem can be applied, summarize your work in a short explanation.
If you get that f(a) isn't defined or the limit does not exist (DNE) then this function isn't continuous at that x-value.
1) Although you might get two different numbers for f(a) and the limit, the numbers HAVE TO BE EQUAL
f(a) is defined
If f(x) has a hole and/or a vertical asymptote, f(x) is not continuous at x=a
f(x) has no asymptotes at x=a
f(x) has no holes at x=a
What's the importance of this?
If one of these main ideas are not met, we know that there is some inconsistencies in the graph (discontinuities).
This determines a whole world of things like for instance, if we can find the derivative at that certain x-value (differentiability implies continuity).
If all three of the main ideas are met, f(x) is continuous at this x-value (x=a).
The limit of f(x) as x approaches a exists
3) The limits from the left and right equal each other.
2) The limit of f(x) as x approaches a FROM THE RIGHT exists.
1) The limit of f(x) as x approaches a FROM THE LEFT exists.
