Advanced High-School Mathematics

SECTION 5.1 Quick Survey of Limits 257

Now letf be a continuous function defined for all real numbers

and compute

Tlim→ 0

1

T

∫∞

−∞Π

Çx−a

T

å

f(x)dx

in terms off anda.

6. (The (real) Laplace transform) Let f = f(x) be a function
   defined for x ≥ 0. Define a new function F = F(s), called the
   Laplace transformoffby settingF(s) =

∫∞

0 e

−sxf(x)dx, where

s≥0. Now letf be the function defined by

f(x) =







1 if 0≤x≤ 1

0 ifx > 1.

Compute the Laplace transformF =F(s) explicitly as a function

ofs.

7. Letf(x) = sin 2πx, x≥0. Compute the Laplace transformF =
   F(s) explicitly as a function ofs. (You’ll need to do integration
   by parts twice!)

5.1.3 Indeterminate forms and l’Hˆopital’s rule

Most interesting limits—such as those defining the derivative—are “in-

determinate” in the sense that they are of the form limx→a

f(x)

g(x)

where the

numerator and denominator both tend to 0 (or to∞). Students learn
to compute the derivatives of trigonometric functions only after they
have been shown that the limit

xlim→ 0

sinx

x

= 1.