The discussion revolves around evaluating the limit of the expression (x)(sin(1/x)) as x approaches infinity. Participants are exploring the behavior of the sine function as its argument approaches zero and the implications of multiplying by an unbounded variable.
Some participants have recognized the indeterminate form of [0 * ∞] and are clarifying their understanding of why the limit approaches 1 based on graphical evidence. There is an acknowledgment of the need to take limits to resolve the expression's behavior.
Participants are grappling with the concept of limits involving infinity and the behavior of trigonometric functions near zero. There is a mention of a prior misunderstanding regarding the nature of the limit, which has prompted further exploration of the topic.
No.Michele Nunes said:Homework Statement
lim (x)(sin(1/x))
x->∞
Homework Equations
The Attempt at a Solution
The correct answer is 1, however I do not understand why. I thought that 1/∞ is essentially 0 and sin(0) = 0 so the whole limit would be 0.
Michele Nunes said:When I graphed it, the limit did seem to approach 1 though. I don't understand why it doesn't work out algebraically.
Ohh, I wasn't aware that 0 * ∞ is considered indeterminate form as well, but that's good to know now though, thank you!Mark44 said:No.
You have x becoming unbounded while sin(1/x) is approaching 0. This is the indeterminate form [0 * ∞], meaning that we can't say without taking the limits what will happen.