How can I get interval in this limit?

Question

f(x) = e^x if x>=0

x-1 if 0<x<1

In(x) if x>=1

Q. Use the graph to determine the values of a for which lim f(x) exists.

x-> a

(Enter your answer using interval notation)

I know it doesn't exist the limit at x=0. So I thought x>0 or x<0 but it was wrong. I want to know the answer and the reason for that answer.

Answer 1

I'm going to assume you meant the f(x) was e^x for x < 0.

Then the graph of the function looks like this:

You should be able to see that the limit for f(x) everywhere EXCEPT for x approaching zero (which would be +1 coming in from the left and -1 coming in from the right and since these two are different, the limit does not exist there).

So the values of "a" where the limit does exist are (-∞, 0) U (0, ∞) which is means all real numbers except zero. You had the correct answer put you listed it as an inequality. Interval notation uses rounded brackets, or parenthesis, when the number next to the parenthesis is NOT included and square brackets [ ] when the number next to the bracket IS included. No < or > signs are used. Start with the smallest (the number most left on the number line) and end with the largest number (most right on the number line) and separate the numbers by a comma. If a number is excluded, as in this case, you break the intervals into multiple parts separated by a "union" symbol: "U"