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.
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1 answer

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>∞ 👍
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