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

  • #1
Michele Nunes
42
2

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.
 

Answers and Replies

  • #2
Michele Nunes
42
2
UPDATE: wait never mind, i just realized how you can do it algebraically
 
  • #3
36,424
8,405

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.
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.
 
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Likes Michele Nunes
  • #4
Michele Nunes
42
2
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!
 

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