Page 55 - Calculus Workbook For Dummies

Chapter 4

Nitty-Gritty Limit Problems

In This Chapter
Algebra, schmalgebra
Calculators — taking the easy way out
Making limit sandwiches
Infinity — “Are we there yet?”
Conjugate multiplication — sounds R rated, but it’s strictly PG

n this chapter, you practice two very different methods for solving limit problems: using
Ialgebra and using your calculator. Learning the algebraic techniques are valuable for two
reasons. The first, incredibly important reason is that the mathematics involved in the alge-
braic methods is beautiful, pure, and rigorous; and, second — something so trivial that per-
haps I shouldn’t mention it — you’ll be tested on it. Do I have my priorities straight or what?
The calculator techniques are useful for several reasons: 1) You can solve some limit prob-
lems on your calculator that are either impossible or just very difficult to do with algebra,
2) You can check your algebraic answers with your calculator, and 3) Limit problems can
be solved with a calculator when you’re not required to show your work — like maybe on a
multiple choice test.

But before we get to these two major techniques, how about a little rote learning. A few limits
are a bit tricky to justify or prove, so to make life easier, simply commit them to memory.
Here they are:
limc = c 1
x " a lim x = 0
x " 3
(y = c is a horizontal 1
line, so the limit equals lim x = 0
x " - 3
c regardless of the sinx
arrow-number — the lim x = 1
x " 0
constant after the arrow.) cosx - 1
lim = 0
1 x " 0 x
lim x = 3 x
x " 0 + 1
lim 1 + m = e
c
1 x " 3 x
lim = - 3
x " 0 - x
Solving Limits with Algebra

You can solve limit problems with several algebraic techniques. But your first step should
always be plugging the arrow-number into the limit expression. If you get a number, that’s
the answer. You’re done. You’re also done if plugging in the arrow-number gives you
3 !3
A number or infinity or negative infinity over zero, like , or ; in these cases
the limit does not exist (DNE). 0 0
Zero over infinity; the answer is zero.