Question

Question:
If lim(f(x)/x)=-5 as x approaches 0, then

lim(x^2(f(-1/x^2))) as x approaches infinity is equal to

(a) 5 (b) -5 (c) -infinity (d) 1/5 (e) none of these

The answer key says (a) 5.

So this is what I know:
Since lim(f(x)/x)=-5 as x approaches 0, then
lim(f(x))=-5x as x approaches 0

However, how could this information be used to evaluate the other limit since the other limit is approaching infinity?

If I were to just ignore this issue this is what I come on with:
Since f(x)=-5x then,
lim(f(-1/x^2))=-5(-1/x^2) as x approaches infinity

So I could simplify lim(x^2(f(-1/x^2))) as x approaches infinity to

lim(x^2(-5(-1/x^2))) as x approaches infinity, then cancel out the x^2 on the numerator and denominator and simplify. Then I'm left with

lim(5) as x approaches infinity, which is equal to 5.

However to reiterate my problem is how can I transfer the knowledge I've obtained from the first limit to the second limit since the first limit is approaching 0 and the second limit is approaching infinity?

Thank you in advance for the help.

Answer 1

let z = -1/x^2
as z->0, x->∞
lim(f(x)/x)=-5
lim(f(z)/z) = -5 as z->0
so, lim(f(-1/x^2)/(-1/x^2)) = - lim(-x^2f(-1/x^2)) = 5 as x->∞