Calculus2
JJJJJEHSH
LHospitalsRule1.pdf
L’Hospital’s Rule
� 1, 2 Suppose that
lim x→a
f(x) = 0 lim x→a
g(x) = 0 lim x→a
h(x) = 1 lim x→a
p(x) = ∞ lim x→a
q(x) = ∞
If the limit is indeterminate, state the form. If the limit is not indeterminate, evaluate the limit.
1. Quotients
a) lim x→a
f(x)
g(x) b) lim
x→a
f(x)
p(x)
c) lim x→a
p(x)
f(x) d) lim
x→a
p(x)
q(x)
2. Products
a) lim x→a
[f(x)g(x)] b) lim x→a
[f(x)p(x)]
c) lim x→a
[h(x)p(x)] d) lim x→a
[p(x)q(x)]
� 3–12 Evaluate the limit. L’Hospital’s rule may or may not apply. If you use L’Hospital’s rule, indicate this by writing L above the operative equal sign.
3. lim x→(π/2)+
cos x
1− sin x 4. lim
x→0
x + tan x
sin x
5. lim t→0
e2t −1 t2
6. lim t→∞
ln(ln t)
t
7. lim x→0
cos mx− cos nx x2
8. lim x→0+
√ x ln x
9. lim y→1
( 1
ln y −
1
y −1
) 10. lim
y→∞ y1/y
11. lim x→0
(1 + x) 1/x
12. lim x→0+
(cos x)1/x 2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13. Prove that for any number p > 0,
lim x→∞
ln x
xp = 0
This shows that the logarithmic function grows more slowly than any power function, no matter how small the power.
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Solutions to Selected Problems
1. a) Indeterminate {
0 0
} b) Not indeterminate lim
x→a f(x) p(x)
= 0
c) Not indeterminate lim x→a
p(x) f(x)
= ∞
d) Indeterminate {∞
∞ }
2. a) Not Indeterminate lim x→a
[f(x) ·g(x)] = 0
b) Indeterminate {0 ·∞}
c) Not indeterminate lim x→a
[h(x)p(x)] = ∞
d) Not Indeterminate lim x→a
[p(x)g(x)] = ∞
3. −∞
4. 2
5. ∞
6. 0
7. n2 −m2
2
8. 0
9. 1 2
10. 1
11. e
12. 1 √ e
13. Proof
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