Page 64 - Calculus Workbook For Dummies

48 Part II: Limits and Continuity

10
Consider the following four types of expressions: x , 5 , !, and x . If a limit at infinity
x
x
x
involves a fraction with one of them over another, you can apply a handy little tip.
These four expressions are listed from “smallest” to “biggest.” (This isn’t a true order-
ing; it’s only for problems of this type; and note that the actual numbers don’t matter;
they could just as easily be x , 3 , !, and x .) The limit will equal 0 if you have a
8
x
x
x
“smaller” expression over a “bigger” one, and the limit will equal infinity if you have a
“bigger” expression over a “smaller” one. And this rule is not affected by coefficients.
1000 x $ 100 x x
For example, lim = 0 and lim x = 3. Note, however, that something
$
x " 3 3 ! x x " 3 500 100
like (2x)! can change the ordering.
Q. Find lim x 3 x . Q. lim 100
x " 3 . 1 01 x " 3 5 x - cosx 2
A. The limit is 0. A. The limit is 0.
100
This is an example of a “small” expression lim 2
x " 3 5 x - cosx
over a “big” one, so the answer is 0. Perhaps 100
this result surprises you. You may think = 3
that this fraction will keep getting bigger = 0
and bigger because it seems that no matter
what power 1.01 is raised to, it will never The values of cosx that oscillate indefi-
2
grow very large. And, in fact, if you plug nitely between –1 and 1 are insignificant
1000 into x, the quotient is big — over compared with 5x as x approaches infinity.
x 3 100
47,000. But if you enter x in graphing Or consider the fact that lim = 0
. 1 01 x " 3 5 x - 10
mode and then set both tblStart and ∆tbl to 100 100
1000, the table values show quite convinc- and that 5 x - cosx 2 < 5 x - 10 for large
ingly that the limit is 0. By the time x = 3000, 100
the answer is .00293, and when x = 10,000, values of x. Because 5 x - cosx 2 is always
the answer is 6 10# - 32 . positive for large values of x and less than
something whose limit is 0, it must also
have a limit of 0.

4
3
3
2
5 x - x + 10 3 x + 100 x + 4
17. What’s lim ? Explain your 18. What’s lim ? Explain your
4
4
x
x " 3 2 x + + 3 x " - 3 8 x + 1
answer. answer.
Solve It Solve It