Page 67 - Calculus Workbook For Dummies

51
Chapter 4: Nitty-Gritty Limit Problems

2. Multiply out the part of the fraction containing the conjugate pair (the denominator here).
9 3 +
^ x - h ` xj
= lim
x " 9 ^ 9 - xh
3. Cancel.
= lim - ` xjl
1 3 +
b
x " 9
a - b
Don’t forget that any fraction of the form always equals –1.
b - a
4. Plug in
1 3 +
= - ` 9j
= - 6
x - 5 - 5 5
f lim =
x " 10 x - 10 10
Multiply by conjugate, multiply out, cancel, plug in.
` x - 5 - 5j ` x - 5 + 5j
= lim $
x " 10 ^ x - 10h ` x - 5 + 5j
5 -
^ x - h 5
= lim
x " 10
^ x - 10 ` h x - 5 + 5j
^ x - 10h
= lim
x " 10 x - 10 ` h x - 5 + 5j
^
1
= lim
x - 5 + 5
x " 10
1
=
10 - 5 + 5
1
=
2 5
5
=
10
g lim cosx - 1 = 0
x " 0 x
Did you try multiplying the numerator and denominator by the conjugate of cosx - 1? Gotcha
again! That method doesn’t work here. The answer to this limit is 0, something you just have to
memorize.
1 1
x - 2 1
h lim = -
x " 2 x - 2 4
1. Multiply numerator and denominator by the least common denominator of the little frac-
tions inside the big fraction — that’s 2x.
1 1
x - 2 2 x
= lim $
x " 2 x - 2 2 x
2. Multiply out the numerator.
^ 2 - xh
= lim
x " ^ x - 2 2 ^h xh
2