Math Is Fun Forum

(xlix)

(xlviii) Complete this sequence:

Consider the numerator (ignoring the minus sign outside the fraction). Notice that the terms in

and

vanish so the highest power of

is

. Its coefficient is

. Now look at the denominator. The highest power of

is also

and its coefficient is

. Hence the limit as

is

using the following rule:

If

and

and

then

That's because you're calculating the LCM of the two numbers, and when you're calculating the LCM you want to be as economical as possible with their common prime divisors. In general, suppose

where the

are distinct primes and the

and

are non-negative integers some of which may be zero. Then

where

for

.

For your example,

Therefore

Let the roots be

,

,

, so

. Then:

So (i) and (iii) are true.

We have

so

is true. The others are not necessarily true: e.g.

has complex roots

and

.

Well, let's hope so.

Sorry if I seem to be too whiney about it. Also, sorry I mistyped your name earlier. It really was a genuine typo.

Substitute

into

then rearrange to the form I gave in my hint.

Not that difficult, I hope?

@julianthemath

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