The limit exists, simply, when the limit from the left equals the limit to the right. As seen on the bottom of "Definition from the limit", the limit from the left (1 - 1.9 - 1.99 - 1.999...) was 8, and the limit from the right (3 - 2.1 - 2.01 - 2.001...) was 8 as well, and so, the limit was 8.
When does a limit not exist?
A limit does not exist when the limit from the left does not equal the limit to the right. For example, the limit of the function (-1)^x as x approaches 1.5 does not exist; the limit from the left (-1)^1 = -1 is nothing like the limit from the right (-1)^2 = + 1. Additionally, the limit does not exist if the function diverges - this means that it continues infinitely in one direction. The limit of (1/x)^2 as x approaches 0 tends to positive infinity from both sides; however, the limit still does not exist, as it goes towards infinity and does not approach any value (going back to the car analogy: in this scenario, the car never stops traveling.).