152 CHAPTER 5. THE DERIVATIVE

(a) limx→0+ (1+3x)cot2x

(b) limx→0sinx−x

x2 = 0

(c) limx→0sinx−x

x3

(d) limx→0tan(sinx)−sin(tanx)

x7

(e) limx→0tan(sin2x)−sin(tan2x)

x7

(f) limx→0sin(x2)−sin2(x)

x4

(g) limx→0e−(1/x2)

x

(h) limx→0( 1

x − cot(x))

(i) limx→0cos(sinx)−1

x2

(j) limx→∞

(x2(4x4 +7

)1/2 −2x4)

(k) limx→0cos(x)−cos(4x)

tan(x2)

(l) limx→0arctan(3x)

x

(m) limx→∞

[(x9 +5x6

)1/3 − x3]

4. Suppose you want to have $2000 saved at the end of 5 years. How much moneyshould you place into an account which pays 7% per year compounded continuously?

5. Using a good calculator, find e.06 −(1+ .06

360

)360. Explain why this gives a measure

of the difference between compounding continuously and compounding daily.

6. You know limx→∞ lnx = ∞. Show that if α > 0, then limx→∞lnxxα = 0.

7. Consider the following function 4

f (x) =

 e−1/x2for x ̸= 0

0 for x = 0

Show that f (k) (0) = 0 for all k so the power series approximations for this functionare all of the form ∑

mk=0 0xk but the function is not identically equal to 0 on any

interval containing 0. Thus this function has all derivatives at 0 and at every other

point, yet fails to be approximated by finite sums of the form ∑mk=0

f (k)(0)k! xk. This

is an example of a smooth function which is not analytic. (Roughly speaking, afunction is analytic when its power series just described approximates the functionnear the point at which all the derivatives are evaluated.) It is smooth because all

derivatives exist and are continuous. It fails to be analytic because ∑mk=0

f (k)(0)k! xk

fails to approximate the function at any nonzero point because it always gives 0 nomatter how large an m is chosen and yet e−1/x2 ̸= 0 if x ̸= 0. In fact, there is asequence of polynomials which will approximate this function on an interval [0,1],but they are not obtained in the way just described as partial sums of a power series.See Section 4.10.

5.20 VideosDerivative of Inverse antiderivatives and integrals

4Surprisingly, this function is very important to those who use modern techniques to study differential equa-tions. One needs to consider test functions which have the property they have infinitely many derivatives butvanish outside of some interval. The theory of complex variables can be used to show there are no examples ofsuch functions if they have a valid power series expansion. It even becomes a little questionable whether suchstrange functions even exist at all. Nevertheless, they do, there are enough of them, and it is this very examplewhich is used to show this.