Ways to find limits (summary):
- If the function is continuous at the value x approaches, then substitute that value and the number you get will be the limit.
- If the function is not continuous at the value x approaches, then
- If you get something that is not zero divided by zero, the limit does not exist (DNE) or equals infinity (see below).
- If you get or the limit may exist. Simplify by factoring, or using different trig functions. Later in the year a method called L’Hôpital’s Rule can often be used in this situation.
- Dominance is a quick way of finding many limits. Exponentials dominate, polynomials, polynomials dominate logarithms, higher powers dominate lower powers. The next post will give some hints about dominance.
Infinity is not a number, but it often is used as if it were. When we say a limit is infinity, what we mean is that the function increases without bound, or there is some x-value that will make the expression larger than any number you choose. Writing things like are common mistakes.
DNE or Infinity? does not exist, and DNE is a correct answer. However, it is a bit better to say the limit is (equals) infinity, indicating that the expression gets larger without bound as x approaches 3. Both answers will get credit on an AP exam. DNE since the one-sided limits (from the left and from the right) are different. Only DNE gets credit here.
Take a look at this AP question 1998 AB-2: In (a) students found that , in (b) they found the minimum value of is and in (c) they had to state the range of the function is . Thus making the students show they knew that this kind of DNE is the kind where the value increases without bound.