- #1

madah12

- 326

- 1

## Homework Statement

I want to prove that a polynomial f(x) and a polynomial g(x) with degrees of k,n where k,n are positive even integer, n>k

that limit x-> - infinity of f(x)-g(x)=-infinity

## Homework Equations

a polynomial can be written as

a

_{1}x^n+a

_{2}x^(n-1)...+a

_{(n-1)}x+a

_{n}

## The Attempt at a Solution

Since n>k and k,n are positive even integers there is a positive even integer C such that

n=(k+C)

therefore we write f(x)=a

_{1}x^k+a

_{2}x^(k-1)+...+a

_{(k-1)}x+a

_{k}/b

_{1}x^(c+k)

g(x)=b

_{1}x^(k+C)+b

_{2}x^(k+C)-1+...+b

_{(k+c-1)}x+b

_{(k+c)}

f(x)-g(x)=[a

_{1}x^k+a

_{2}x^(k-1)+...+a

_{(k-1)}x+a

_{k}]-[b

_{1}x^(k+C)+b

_{2}x^((k+C)-1)+...+b

_{(k+c-1)}x+b

_{(k+c)}]

=b

_{1}x^(k+c)[[a

_{1}/(b

_{1}*x^c) + a

_{2}/(b1*x^(-c-1)) ...a

_{(k-1)}/b

_{1}x^(c+k-1) + a

_{k}/b

_{1}x^(c+k)] -[ 1+b

_{2}/b

_{1}x +...+b

_{(c+k-1)}/b

_{1}x^(k+c-1)+b

_{c+k}/b

_{1}x^(c+k)]]

as x->-infinity

c/x^n = 0 THEOREM

therefore

since k+c is even then lim x->-infinity x^(c+K)=infinity

so limit x->-infinity f(x)-g(x)

=infinity *[(0+0+...+0+0)-(1+0...+0+0)=infinity * -1 = -infinity.

I know this isn't any kind of rigorous proof so how can I improve it?