Calculus I Name:
TA/classroom: Student ID:
Quiz 7
Dec. 5, 2007 1. (8 pts) Estimate √
4.02 by the method of linear approximation.
Let f (x) = √
x, f0(x) = 12x−1/2. Since f (4) = 2 and 4 ≈ 4.02, we are looking for a linear approxmiation near x = 4.
L(x) = f (4) + f0(4)(x− 4) = 2 +1212(x− 4) = 2 +14(x− 4).
So L(4.02) = 2 + 14(4.02− 4) = 2.005 ≈√ 4.02 2. (6 pts) Compute
xlim→0
ex− 1 x2
Note that limx→0ex− 1 = 0 and limx→0x2 = 0. The limit has an indeterminate form
0
0 and we can apply the L’Hˆopital’s Rule.
xlim→0
ex− 1 x2 = lim
x→0
ex
2x = DN E Note that limx→0− ex
2x =−∞ and limx→0+ ex 2x =∞.
3. (6 pts) Compute
x→0lim+xx
Note that the limit has an indeterminate form 00 and we can apply the L’Hˆopital’s Rule. Let y = xx, so that ln y = ln xx = x ln x (x > 0). Now consider the limit
lim
x→0+ln y = lim
x→0+x ln x
= lim
x→0+
ln x 1/x
= lim
x→0+
1/x
−1/x2 (By L’Hˆopital’s Rule.)
= lim
x→0+−x
= 0 Thus
lim
x→0+xx = lim
x→0+eln y
= elimx→0+ln y (ex is a continuous function.)
= e0
= 1
