If f has domain [0, infinity) and has no horizontal asymptotes, then lim_(x->infinity) f(x) = infinity or lim_(x->infinity) f(x) = -infinity.

Can someone please clarify whether this statement is true or false?

Thank you

To determine whether the statement is true or false, we need to consider the given conditions and apply the definition of a limit.

The statement "If f has domain [0, infinity) and has no horizontal asymptotes, then lim_(x->infinity) f(x) = infinity or lim_(x->infinity) f(x) = -infinity" is actually false.

Here's why:

1. If a function f has no horizontal asymptotes, it means that as x approaches infinity, the function does not approach a finite value. This implies that the function either approaches positive infinity, negative infinity, or it oscillates indefinitely.

2. However, based on the given domain of f as [0, infinity), we know that the function is defined only for x values greater than or equal to 0. This means that for any x such that x < 0, the function is undefined. Consequently, when taking the limit as x approaches infinity, we are only considering values where x is greater than or equal to 0.

3. Since the function is undefined for x < 0, it is not possible to make any conclusions about the behavior of the function as x approaches infinity, solely based on the given domain [0, infinity). The function may have different behaviors for x < 0, which can have an impact on its overall behavior as x approaches infinity.

Therefore, without further information about the behavior of f for x < 0, we cannot definitively say that lim_(x->infinity) f(x) equals either positive infinity, negative infinity, or any specific value. The statement is false.

The statement is false.

If a function has no horizontal asymptotes, it means that the function does not approach a specific value as x approaches positive or negative infinity. In other words, the limit as x approaches infinity may still exist even if there are no horizontal asymptotes.

For a function f with domain [0, infinity), the limit as x approaches infinity can be any real number, infinity, or negative infinity. It is not necessary that the limit will be infinity or negative infinity. Therefore, the statement stating that lim_(x->infinity) f(x) = infinity or lim_(x->infinity) f(x) = -infinity is not always true.

Examine whether f(x)=sin(x) satisfies the requirements for f(x), and decide if the statement is true or false.