5.19. EXERCISES 151
Example 5.18.1 Suppose you have $100 and you put it in a savings account which pays6% compounded continuously. How much will you have at the end of 4 years?
From the above discussion, this would be 100e(.06)4 = 127.12. Thus, in 4 years, youwould gain interest of about $27.
5.19 Exercises1. Find the limits.
(a) limx→03x−4sin3x
tan3x
(b) limx→ π2 − (tanx)x−(π/2)
(c) limx→1arctan(4x−4)arcsin(4x−4)
(d) limx→0arctan3x−3x
x3
(e) limx→0+9secx−1−13secx−1−1
(f) limx→03x+sin4x
tan2x
(g) limx→π/2ln(sinx)x−(π/2)
(h) limx→0cosh2x−1
x2
(i) limx→0−arctanx+x
x3
(j) limx→0x8 sin 1
xsin3x
(k) limx→∞ (1+5x)2x
(l) limx→0−2x+3sinx
x
(m) limx→1ln(cos(x−1))
(x−1)2
(n) limx→0+ sin1x x
(o) limx→0 (csc5x− cot5x)
(p) limx→0+3sinx−12sinx−1
(q) limx→0+ (4x)x2
(r) limx→∞x10
(1.01)x
(s) limx→0 (cos4x)(1/x2)
2. Find the following limits.
(a) limx→0+1−
√cos2x
sin4(4√
x).
(b) limx→02x2−25x
sin(
x25
)−sin(3x)
.
(c) limn→∞ n( n√
7−1).
(d) limx→∞
( 3x+25x−9
)x2.
(e) limx→∞
( 3x+25x−9
)1/x.
(f) limn→∞
(cos 2x√
n
)n.
(g) limn→∞
(cos 2x√
5n
)n.
(h) limx→3xx−27x−3 .
(i) limn→∞ cos(
π
√4n2+13n
n
).
(j) limx→∞
3√x3 +7x2
−√
x2 −11x
.
(k) limx→∞
5√x5 +7x4
− 3√x3 −11x2
.
(l) limx→∞
(5x2+72x2−11
) x1−x
.
(m) limx→∞
(5x2+72x2−11
) x lnx1−x
.
(n) limx→0+ln(
e2x2+7
√x)
sinh(√
x) .
(o) limx→0+7√x− 5√x9√x− 11√x
.
3. Find the following limits.