Page 56 - Calculus Workbook For Dummies

40 Part II: Limits and Continuity

0
When plugging in fails because it gives you , you’ve got a real limit problem, and you
0
have to convert the fraction into some expression where plugging in does work. Here
are some algebraic methods you can try:

FOILing
Factoring
Finding the least common denominator
Canceling
Simplification
Conjugate multiplication

The following examples each use a different method to solve the limit.

Q. Evaluate lim 16 - x . Q. What’s lim x - - 6 ?
2
x
2
x
x " 16 4 - x x " - 2 x + - 2
A. The limit is 8. A. The limit is 5 .
3
1. Try plugging 16 into x — no good. 1. Try plugging –2 into x — that gives
0
2. Multiply numerator and denomina- you , so on to plan B.
tor by the conjugate of 4 - x, 0
2. Factor and cancel.
namely 4 + x.
2 ^
The conjugate of a two-term expres- = lim ^ x + h x - 3h
2 ^
sion has a plus sign instead of a x " - ^ x + h x - 1h
2
minus sign — or vice-versa. ^ x - 3h
= lim
x " - ^ x - 1h
2
^ 16 - xh ` 4 + xj
lim $ 3. Cancel now and plug in.
x " 16 4 - xj 4 + xj
` `
- 2 - 3
=
3. FOIL the conjugates and simplify. - 2 - 1
- 5
=
- 3
x 4 +
^ 16 - h ` xj Because,
= lim = 5
x " 16 2 2 of course, 3
c 4 - x m
^ a - h ^ b =
b a + h
2
x 4 +
^ 16 - h ` xj a - b 2 .
= lim
x " 16 ^ 16 - xh
= lim 4 + xj
`
x " 16
4. Now you can cancel and then
plug in.
= 4 + 16
= 8
Note that while plugging in did not work
in Step 1, it did work in the final step.
That’s your goal: to change the original
expression — usually by canceling — so
that plugging in works.