True or False,... Im confused
If the continuous function f(x) has a domain (-∞, +∞), then either lim --> ∞ exists, or limit X--> ∞ is +-∞
I do not see how it could be otherwise. If there were a singularity at infinity resulting in an undefined limit, then how could the domain include plus and minus infinity?
False.
To determine the correctness of this statement, we need to analyze it step by step.
The statement asserts that if a continuous function f(x) has a domain of (-∞, +∞), then one of two conditions must be true: either the limit as x approaches infinity exists, or the limit as x approaches infinity is positive or negative infinity.
To evaluate this claim, we can consider an example where f(x) = x^2. This function is continuous and its domain is indeed (-∞, +∞). However, if we compute the limit as x approaches infinity in this case, we get:
lim(x->∞) x^2 = +∞
This does not align with the statement, which claimed that the limit as x approaches infinity can either exist or be positive or negative infinity. In this case, the limit exists and is positive infinity.
Therefore, since we have found a counterexample that shows not all continuous functions with a domain of (-∞, +∞) will have a limit as x approaches infinity that corresponds to either of the conditions stated, the statement is false.