12.1 Logic
12.2 Sequences
12.3 Limits
12.4 Imaginary and Complex Numbers
12.5 Key Formulas
12.6 Review Questions
12.7 Explanations
For a function f(x), the limit of the function is the value that the function approaches as x approaches a certain number. Here’s an example:
What is the limit of f(x) = (x +2)/(x2 – 2x – 8) as x approaches –2?
Normally, finding the limit should be easy. You would simply plug the value that x approaches, in this case –2, into the function and produce the limit. But the Math IC typically asks only one type of question about limits—the limit of a function at a point at which the function is not defined. Try and plug –2 into the function.
It seems that the function is not defined at x = –2, because division by zero is not allowed. This is precisely what the test wants you to think. But the assumption that this function is undefined at –2 is incorrect. Luckily, there is an easy way to solve for the limit.
First, you need to factor the function so that it is in its most simplified state. For our example, the denominator can be factored:
Once the denominator has been factored, it’s easy to see that the function x+2(x+2)(x–4) simplifies to 1x–4. This simplified fraction can be evaluated at x = –2:
1–6 is the limit of f(x) at x = –2.
You should choose “the limit does not exist at this point” answer only if a function is undefined at the point at which you wish to find a limit and the function cannot be factored any further.
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