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50 2. Introductory applications

it is left as an exercise to confirm that these results can be obtained directly from
the familiar solution of the quadratic equation, suitably approximated (by using the
binomial expansion) for (Similar problems based on quadratic equations can
be found in exercise Q2.1.)
This simple introductory example covers the essentials of the technique:
find all the different asymptotic forms of and investigate if roots
exist for each (dominant) asymptotic representation. Let us now apply this to a slightly
more difficult equation which, nevertheless, has a similar structure.

E2.1 A cubic equation
We are to find approximations to all the real roots of the cubic equation

for First, for x = O(1), we have

and this approximation admits the roots x = ±1; a better approximation is then ob-
tained by writing

so that we obtain

This equation requires that and so on; two roots are therefore

Now the ‘asymptotic expansion’

remains valid as but not as it breaks down where or
We write and then

or