Limit as X approaches ∞ of (X)(sin(1/X))

[Michele Nunes]

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. When I graphed it, the limit did seem to approach 1 though. I don't understand why it doesn't work out algebraically.

 

[Michele Nunes]

UPDATE: wait never mind, i just realized how you can do it algebraically

 

[Mark44]

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.

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.

When I graphed it, the limit did seem to approach 1 though. I don't understand why it doesn't work out algebraically.

 

[Michele Nunes]

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.

Ohh, I wasn't aware that 0 * ∞ is considered indeterminate form as well, but that's good to know now though, thank you!