Thread: Limits (With a specified range?)

Suppose 0≤ f(x) ≤1 for all x, find

lim x⁴ f(x)
x→0.

This is exactly how the question is presented in my manual and I am not sure if it is trying to say that the function is x⁴ or something else. Also, I am not sure what relevance the first part of the question has to finding the limit. I believe it is representing the range, although I don't know what that means as far as limits go. Thank-you for your help.

The function can't be , because isn't surely bounded by for all , but if you rewrite:
and you know then the limit is ...

Thank-you for your quick reply.

I am still a little unclear as to the part which states:

0≤ f(x) ≤1 for all x.

Does this represent the range of f(x)?

I understand that if the limit of
lim x⁴
x→0
is 0 that multiplying this by the limit of f(x) will yield a limit of 0. I am just wondering about the relevance of the range stated and whether that should affect my solution. Thank-you.

Originally Posted by Pewter12

0≤ f(x) ≤1 for all x.
Does this represent the range of f(x)?
I understand that if the limit of
lim x⁴x→0 is 0 that multiplying this by the limit of f(x) will yield a limit of 0.

Actually this is the relevance of that condition.
implies that .
From that it follows at once that the limit is zero.

We do not know that

I'm sorry, what I meant was that the product of

lim x⁴ x lim f(x)
x→0 x→0

would be 0 because

lim x⁴
x→0 = 0

Is there a particular rule telling us that
0≤ f(x) ≤1

implies also that

0≤ x⁴f(x) ≤1?

Originally Posted by Pewter12

Is there a particular rule telling us that
0≤ f(x) ≤1
implies also that 0≤ x⁴f(x) ≤1?

Well I did not say that it did: .
However, if then .
So in the limit process that does in fact follow.

But my point in post #4 is that you use the squeeze play.