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UAE_Math_Grade_12_Vol_1_SE_718383_ch3
GO01962-Smith-v1.cls
Again, we have not yet found the original limit. However,
0
lim y = lim e ln y = e = 1,
x→0 + x→0 +
which is consistent with Figure 4.22.
y
EXAMPLE 2.10 The Indeterminate Form ∞ 0
4
Evaluate lim (x + 1) 2∕x .
x→∞
3
0
Solution This limit has the indeterminate form (∞ ). From the graph in Figure
2 4.23, it appears that the function tends to a limit around 1 as x → ∞. We let
y = (x + 1) 2∕x and consider
1
[ 2 ]
x lim ln y = lim ln (x + 1) 2∕x = lim ln (x + 1) (0 ⋅ ∞)
20 40 60 80 100 x→∞ x→∞ x→∞ x
FIGURE 4.23 = lim 2 ln (x + 1) ( ∞ )
y = (x + 1) 2∕x x→∞ x ∞
d [2 ln (x + 1)]
dx 2(x + 1) −1
= lim = lim By l’Ĥopital’s Rule.
x→∞ d x x→∞ 1
dx
2
= lim = 0.
x→∞ x + 1
0
We now have that lim y = lim e ln y = e = 1,
x→∞ x→∞
as expected.

4
EXERCISES .2

WRITING EXERCISES other. If f(t) and g(t) represent the positions of the runners
at time t ≥ 0, explain why we can assume that f(0) = g(0) =
′
1. L’Ĥopital’s Rule states that, in certain situations, the ratios 0 and lim f (t) = 2. Explain in terms of the runners’ posi-
′
of function values approach the same limits as the ratios t→0 + g (t)
of corresponding derivatives (rates of change). Graphically, tions why l’Ĥopital’s Rule holds: that is, lim f(t) = 2.
this may be hard to understand. To get a handle on this, t→0 g(t)
f(x)
consider where both f(x) = ax + b and g(x) = cx + d are
g(x)
f(x) In exercises 1–40, find the indicated limits.
linear functions. Explain why the value of lim should
x→∞ g(x) x + 2 x − 4
2
depend on the relative sizes of the slopes of the lines; that 1. lim 2 2. lim 2
x→2 x − 3x + 2
x→−2 x − 4
′
f (x)
is, it should be equal to lim . 3x + 2 x + 1
2
x→∞ g (x) 3. lim 4. lim
′
x→∞ x − 4 x→−∞ x + 4x + 3
2
2
2. Think of a limit of 0 as actually meaning “getting e − 1 sin t
2t
very small” and a limit of ∞ as meaning “getting very 5. lim 6. lim 3t
large.” Discuss whether the following limit forms are t→0 t t→0 e − 1
indeterminate or not and explain your answer: ∞−∞, tan −1 t sin t
∞
1 , 0 ⋅ ∞, ∞ ⋅ ∞, ∞ , 0 and 0 . 7. lim 8. lim −1
0
0
t→0 sin t
t→0
0 sin t
3. A friend is struggling with l’Ĥopital’s Rule. When asked to 9. lim sin 2x 10. lim cos −1 x
Copyright © McGraw-Hill Education 4. Suppose that two runners begin a race from the starting 11. lim sin x − x 12. lim tan x − x
work a problem, your friend says, “First, I plug in for x and
x→ sin x
x→−1 x − 1
2
get 0 over 0. Then I use the quotient rule to take the deriva-
tive. Then I plug x back in.” Explain to your friend what the
x
x
3
3
x→0
x→0
mistake is and how to correct it.
√
t − 1
ln t
14. lim
13. lim
t→1 t − 1
t→1 t − 1
line, with one runner initially going twice as fast as the
247