Can you guys help me with my home work? A random polynomial P(x)

X 2 is supused to be square x

And in the last lim of the pic it's supposed to tends 2 not infinite

The numerator has a 4x^2 term

>>8431710
Why?

>>8431721
Because of the limit to infinity. It means that at infinity, the limit is equivalent to 4x^2 / x^2 which then is equal to 4.

Not only do you know it has a 4x^2 term, you also know that is the leading term because if it has a bigger term (like an x cubed) then the limit would diverge to infinity.

>>8431696
You can factor the denominator into (x-2)(x+2), what does that mean for the convergence at x=2 and P(x)?

>>8431732
I somehow got to simplify x-2

>>8431727
Oh thanks my dude now i undrstand

>>8431732
It should be like something*(x-2) and that something for P(2) should be 9

Lim x -> inf of P(x) / (x^2 - 4) = K

Apply Lhopitals rule twice and reason P(x) has to be polynomial of degree 2.

P(x) = Ax^2 + Bx + C
P'(x) = 2Ax + B
P''(x) = 2A

x^2 - 4
2×
2

2A/2 = K

A = K

SO P(x) = Kx^2 + Bx + C where B and C can be any number you want.

>>8431752
I know it's degree is 2 by now but i am asked what's the polynomial P(x)

>>8431696
>Can you guys help me with my home work?
no

P(x) = Kx^2 + Bx + C where B and C can be any number you want.

K is the number the limit is equal to.

Ok thx guys you are the best i've solved it