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42 1. Mathematical preliminaries

(e) with for
(f) repeat (e) for
(g) with
(h) with
[Hint: seek a solution
(i) with
(j) repeat (i) for
(k) with for « = 1;
(1) repeat (k) for n = 1/2 and
(m) repeat (k) for n = 2 and
(n) repeat (k) for n = 3 and
Q1.4 Uniform or non-uniform? Is the behaviour of each of these functions uniformly
valid as and (To answer this, compare followed by
with the reversed order of taking the limits.)

(a) (b) (c) (d)

Q1.5 Limits involving exp(x) and ln(x). (a) For x > 1 and show that 0 <
and hence that (b) Interpret these as
inequalities for lnx, choose and show that
(c) Now write x = 1/y and hence as
(d) In (b), write and show that
as (e) In (d), write x = exp(y) and deduce that
for
Q1.6 Examples of O, o and ~. Determine which of the following are correct as

(a) (b)
(c) (d)
(e) (f)
(g) (h)
(i) (j)
(k)

Q1.7 Dominant behaviour. Find the dominant behaviour, i.e. find g (x) so that
as for each of these functions, f(x).

(a) (b) (c)

(d) (e) (f)
(g) F[G(x)] where