Page 69 - Calculus Workbook For Dummies

53
Chapter 4: Nitty-Gritty Limit Problems

4. Substitute u for 3x. And, because u approaches 0 as x approaches 0, you can substitute u
for x under the lim symbol.
1 u
= lim
3 u " 0 sinu
1
= 1 $
3
1
=
3
sinx sinx x
Because lim x = 1, the limit of the reciprocal of x , namely , must equal the
x " 0 sinx
reciprocal of 1 — which is, of course, 1.
*l lim x = 1
x " 0 tanx
sinx sinx
1. Use the fact that lim x = 1 and replace tanx with cosx .
x " 0
x
= lim
x " 0 sinx
cosx
2. Multiply numerator and denominator by cosx.
x cosx
= lim $ cosx
x " 0 sinx
cosx
x cosx
= lim
x " 0 sinx
3. Rewrite the expression as the product of two functions.
x cosx
= limc $ m
x " 0 sinx 1
^
4. Break this into two limits, using the fact that lim f x $ ^` ^ h g xhj = lim f x $ h lim g xh
^
x " c x " c x " c
(provided that both limits on the right exist).
x
= lim $ lim cosx
x " 0 sinx x " 0
= $ 1
1 1 =
2
m lim x - 5 x - 24 = –11
x " - 3 x + 3
You want the limit as x approaches –3, so pick a number really close to –3 like –3.0001, plug that
2
x - 5 x - 24
into x in your function and enter that into your calculator. (If you’ve got a calcula-
x + 3
tor like a Texas Instruments TI-83, TI-86, or TI-89, a good way to do this is to use the STO→
2
x - 5 x - 24
button to store –3.0001 into x, then enter into the home screen and punch enter.)
x + 3
The calculator’s answer is –11.0001. Because this is near the round number –11, your answer
is –11. By the way, you can do this problem easily with algebra as well.
n lim sinx = 1
1
-
x " 0 tan x
sinx
Enter the function in graphing mode like this: y = - 1 . Then go to table setup and enter a
tan x
small increment into ∆tbl (try 0.01 for this problem), and enter the arrow-number, 0, into
tblStart. When you scroll through the table near x = 0, you’ll see the y values getting closer and
closer to the round number 1. That’s your answer. This problem, unlike problem 13, is not easy
to do with algebra.