Ratio of functions, surjective (analysis course)

[riskandar]

Homework Statement

let f: R->R be a continuous function
Suppose k>=1 is an integer such that

lim f(x)/x^k = lim f(x)/x^k = 0
x->inf x->-inf

set g(x)= x^k + f(x)

g: R->R

Prove that
(i) if k is odd, then g is surjective
(ii) if k is even, then there is a real number y such that the image of g is [y,inf)

Homework Equations

The Attempt at a Solution

I am completely stuck at this all I can think of is x^k goes to infinity then the ratio of the functions can go to 0 if either f(x) goes to 0 or f(x) is a constant or f(x) goes to infinity slower than x^k (I am not sure about this)

Any help will be very much appreciateve

 

[ystael]

Because $$g$$ is continuous, you can use the intermediate value theorem. To show that $$g$$ is surjective, it is enough to show that $$g$$ becomes both very large and very large negative.

To prove this part, you need to think about the qualitative behavior of $$f$$ and $$g$$. The hypothesis on $$f$$ says that $$f(x)$$ is negligible compared to $$x^k$$ as $$|x|$$ becomes large. Therefore $$g(x)$$ should "behave almost like" $$x^k$$ as $$|x|$$ becomes large. Figure out a way to make this precise.

 

[riskandar]

Because $$g$$ is continuous, you can use the intermediate value theorem. To show that $$g$$ is surjective, it is enough to show that $$g$$ becomes both very large and very large negative.

To prove this part, you need to think about the qualitative behavior of $$f$$ and $$g$$. The hypothesis on $$f$$ says that $$f(x)$$ is negligible compared to $$x^k$$ as $$|x|$$ becomes large. Therefore $$g(x)$$ should "behave almost like" $$x^k$$ as $$|x|$$ becomes large. Figure out a way to make this precise.

Thank you for the reply.
Where do I use the intermediate value theorem? Is it to prove surjective?

 

[ystael]

Thank you for the reply.
Where do I use the intermediate value theorem? Is it to prove surjective?

Yes.