- #1
gymko
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Homework Statement
Why is it?
lim{x-> infinity} f(x)/x = infinity => lim{x-> infinity} f(x) = infinity
What is evidence?
Thank you very much.
The limit of f(x)/x as x approaches infinity equals infinity because as x gets larger and larger, the value of f(x) also gets larger. Since the denominator is approaching infinity, the quotient will also approach infinity.
The limit of f(x) as x approaches infinity is directly related to the limit of f(x)/x as x approaches infinity. In fact, the limit of f(x)/x can be found by dividing the limit of f(x) by the limit of x. This is because as x gets larger, the value of x becomes insignificant compared to the value of f(x).
No, the limit of f(x) as x approaches infinity cannot be equal to infinity if the limit of f(x)/x as x approaches infinity does not exist. This is because if the limit of f(x)/x does not exist, it means that the quotient does not approach a specific value as x gets larger. Therefore, the limit of f(x) cannot approach a specific value, and thus cannot equal infinity.
Yes, there are cases where the limit of f(x) as x approaches infinity can equal infinity even if the limit of f(x)/x as x approaches infinity is not infinity. This can happen if the limit of f(x)/x approaches infinity from one side and negative infinity from the other side. In this case, the limit of f(x) would still be equal to infinity as x approaches infinity, even though the limit of f(x)/x does not exist.
Yes, it is possible for the limit of f(x) as x approaches infinity to equal a value other than infinity. This can happen if the limit of f(x)/x approaches a finite value or if the limit of f(x)/x approaches zero. In these cases, the limit of f(x) would be equal to the same value as the limit of f(x)/x multiplied by infinity or zero, respectively.