142 Functions, limits, derivatives
23 Sometime we will learn that(1) lim 0.
,.2-Hence there must be an integer N such that(2) T.n3 I
< 100when n > N. Some numerical calculations can make us quite sure that (2) is
valid when n > 20. Even though the author considers the problem to be too
difficult for assignment at this time, it may be worthwhile to seek a way to deter-
mine whether (2) is valid when n > 20.
24 Prove that if x is a rational number, say p/q, where p and q are integers,
then sin n!irx = 0 for each sufficiently great integer n. Prove that if x is an
irrational number, then sin n!hrx 0 0 for each integer n. Using these results,
show that
1 - lim sgn sin2 narx = D(x)
n- w
where D is the dizzy dancer function for which D(x) = 1 when x is rational and
D(x) = 0 when x is irrational.
25 Some old analytic geometry books pretend to prove that if n is a positive
integer and Po, P,, , P. are polynomials in x, then the line having the
equation x = x, will be an asymptote of the graph of the equation(1) Po(x)yn + Pl(x)yn-1 +.... + P,_i(x)y + Pn(x) = 0
provided Po(x1) = 0. These old books present unclear and unreliable treatments
of matters involving limits and asymptotes, however, and the stated result is
false. Prove that the line having the equation x = 0 is not an asymptote of the
graph of the equation(2) x2y2+x2y+1 =0.
Remark: An example which establishes falsity of an assertion is called a counter-
example. Persons who speak German (and many others also) call ita Gegen-
beispiel. The simpler equation x2y2 + 1 = 0 serves the present purpose; the
graph of this equation is the empty set.(^26) Prove that if f1, f2, f3 are continuous at a, if
(1) lim y(x) = 00,
z- a+
and if, for some positive number 8,
(2) f1(x)[y(x)]2 +f2(X)Y(X) +f3(x) = 0 (a < x < a + S),
then f1(a) = 0. Hint: Choose a positive number S1 such that S1 < S and y(x) >
I when a < x < a + Si. Then, supposing that a < x < a + S1, divide the
members of (2) by [y(x)]2 to obtain
fl\x) +f2(x)+ fs(x) = 0.
y(x) [y(x))2