Explain whether the statement is true and give an explanation or a counterexample:
If lim f(x) = ∞ and lim g(x) = ∞ , then lim (f(x)-g(x)) = 0.
x->∞ x->∞ x->∞
I think the answer is false, but I'm not sure how to go about explaining why.
∞-∞ is undefined
consider
x^3-x^2 --> ∞
time to read up on limits and infinity. It's a tricky subject.
To determine the veracity of the statement, we can apply limit arithmetic rules and analyze the behaviors of f(x) and g(x) individually.
The statement claims that if the limits of f(x) and g(x) as x approaches infinity both converge to infinity, then the limit of (f(x) - g(x)) as x approaches infinity converges to zero.
Let's consider a counterexample to disprove the statement. Suppose we have two functions f(x) = 2x and g(x) = x. In this case, as x approaches infinity, both f(x) and g(x) tend to infinity.
Now, let's find the limit of (f(x) - g(x)) as x approaches infinity:
lim (f(x) - g(x)) = lim (2x - x) = lim x = ∞.
As we can see, the limit of (f(x) - g(x)) is not 0; instead, it is infinity. Therefore, the statement is false, and we have provided a counterexample that demonstrates why it is incorrect.
This counterexample illustrates that in general, if the individual limits of f(x) and g(x) are both infinity, the limit of their difference ((f(x) - g(x))) may be indeterminate or infinity, but it is not necessarily 0.