Limit Problem: Itermediate Value Theorem

[MrBailey]

Hello out there. I hope everyone is doing well.
I could use a little guidance on this:

suppose $$f$$ is continuous for all $$x$$, and

$$\lim_{x\rightarrow -\infty}f(x) = -1$$ and $$\lim_{x\rightarrow +\infty}f(x) = 10$$

Show that $$f(x) = 0$$ for at least one $$x$$

I know I need to use the Intermediate Value Theorem and the definition of the limit...but I'm not really sure how to apply them.

Thanks,
Bailey

 

[benorin]

The definition of a limit at infinity will give you some finite value of x for which f(x) is within some neighborhood of 10 (and, separately, -1). Then use IVT.

 

[MrBailey]

thanks!

I see it now.

Bailey

 

[mathwonk]

try to prove then that every odd degree polynomial has a root.

 

[benorin]

Nice addition.

 

[mathwonk]

it was homework problem in frosh calc that i did not get at the time.