Page 62 - Calculus Workbook For Dummies

46 Part II: Limits and Continuity

Making Yourself a Limit Sandwich

The sandwich or squeeze method is something you can try when you can’t solve a limit
problem with algebra. The basic idea is to find one function that’s always greater than
the limit function (at least near the arrow-number) and another function that’s always
less than the limit function. Both of your new functions must have the same limit as x
approaches the arrow-number. Then, because the limit function is “sandwiched”
between the other two, like salami between slices of bread, it must have that same
limit as well. See Figure 4-1.

y
f
f 5
g 4
Figure 4-1: 3 g
A limit 2
sandwich —
functions f h 1 h
and h are x
the bread -4 -3 -2 -1 1 2 3 4 5
and g is the -1
salami. -2

Q. What’s lim x ? x = 0) so you can use the simple
x
x " 0 3 function y = 0 as the bottom slice of
A. The limit is 0. bread. Of course, it’s obvious that
lim0 = 0. Finding a function for the
x "
0
1. Try plugging in 0. No good, you get top slice is harder. But let’s just say
0 over 0.
that for some mysterious reason,
You should be able to solve this you know that y = x is greater
limit problem with algebra. But let’s than x near the arrow-number —
say you tried and failed so now 3 x
you’re going to try the sandwich the only place that matters for
method. the sandwich method. Because
lim x = 0, y = x makes a good
2. Graph the function. x " 0
top slice.
Looks like the limit as x approaches
0 is 0. You’re done. Because x is
3 x
3. To prove it, try to find two bread squeezed between y = 0 and
functions that both have a limit of y = x , both of which have limits of
0 as x approaches 0. x
0 as x approaches 0, must also
It’s easy to show that the function is have a limit of 0. 3 x
always positive (except perhaps at