Question

A factory owner estimates that the number of units(in thousands) N of a new product being produced after x months can be estimated by the function N(x) = (4x^2+3x)/(1+x)^2. What happens to the production over time ( x increases indefinitely) ? Explain.

Answer 1

is it like
\[N(x) = \frac{ 4x ^{2} +3x }{ (1+x)^{2} }\]

Answer 2

looks like it... and the question is, what happens to N(x) as x--> infinity

Answer 3

yes

Answer 4

on these, you should first try to simplify if you can. Try factoring on the top...

Answer 5

maybe, I'm actually not sure that gets you very far...

Answer 6

Sorry, instead, expand the bottom by multiplying it out to be x^2 + 2x + 1

Answer 7

So, as x gets "really big".... the x^2 terms get gigantic... and the x terms get big but not gigantic, and the constants stay the same and by comparison get puny

Answer 8

In the end, the only thing that matters is that you have 4x^2 on top and x^2 on bottom, and for giant x, all other terms are meaningless, so the equation reduces to the number 4, since the x^2 on top and bottom cancel

Answer 9

I think all that is correct... would appreciate a correction or confirmation @tamtoan :)

Answer 10

I might have oversimplified for "giant" x values

Answer 11

N(x) = 4x^2 / (1 + x)^2 + 3x/(1 + x)^2 = [2x/(1+x)]^2 + 3x/(1+x)^2

as x get larger and larger, second term will not be negative...and the first term, 2x get bigger faster than 1 + x so that first time will be really big, ...as x get to infinity, N(x) should also go to infinity

Answer 12

How sure are you :) ? I like your answer better than mine... but we can call for backup if you aren't positive. As a matter of real-world problems, a factory owner isn't likely to estimate infinite production per month... more likely that production "settles down" to some long term figure after short term effects dissipate out.

Answer 13

@satellite73 got a minute for another limit question?

Answer 14

i am probably about 80% sure :) haven't touch any kind of math problems for a long while :)
can check with backup and see what happen :)