Solving the Limit of (1-cos x)sin(1/x)

  • Thread starter nietzsche
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In summary, the limit of (1-cos x)sin(1/x) as x approaches 0 is 0. This can be shown using the squeeze theorem by bounding sin(1/x) between -1 and 1 and using the fact that (1-cos x) approaches 0 as x approaches 0.
  • #1
nietzsche
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Homework Statement



Find the following limit:

[tex]
\lim_{x \to 0} (1-\text{cos }x)\text{sin }\frac{1}{x}
[/tex]

Homework Equations





The Attempt at a Solution



(1-cos x) -> 0 as x -> 0. sin (1/x) oscillates infinitely many times as x -> 0.

intuition tells me that the limit is 0, but how do i show that?

some ideas i have are using the fact that |sin(1/x)| =< 1, but I'm not sure.
 
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  • #2
Try the squeeze theorem with something that converges to zero like [tex]\frac{1-\cos{x}}{x}[/tex].
 
  • #3
i ended up doing this.

[tex]
\begin{align*}
-1 &\leq& \text{sin }\frac{1}{x} &\leq& 1\\
-(1-\text{cos }x) &\leq& (1-\text{cos }x)(\text{sin }\frac{1}{x}) &\leq& 1- \text{cos }x
\end{align*}
[/tex]

since both of the terms on the side equal 0 at x=0, by the squeeze theorem, the middle term also goes to 0.
 
  • #4
That's how I would have done it. Well done!
 

1. What is the limit of (1-cos x)sin(1/x) as x approaches 0?

The limit of (1-cos x)sin(1/x) as x approaches 0 is 0.

2. How do you solve the limit of (1-cos x)sin(1/x)?

To solve the limit of (1-cos x)sin(1/x), you can use the Squeeze Theorem or L'Hopital's Rule.

3. Can the limit of (1-cos x)sin(1/x) be evaluated using algebraic manipulation?

No, the limit of (1-cos x)sin(1/x) cannot be evaluated using algebraic manipulation because it involves a trigonometric function and a reciprocal function.

4. Is there a graphical representation of the limit of (1-cos x)sin(1/x)?

Yes, the limit of (1-cos x)sin(1/x) can be graphically represented by a removable discontinuity at x = 0.

5. What is the relationship between (1-cos x)sin(1/x) and the Sine function?

The relationship between (1-cos x)sin(1/x) and the Sine function is that (1-cos x)sin(1/x) is a transformation of the Sine function, where the amplitude is multiplied by (1-cos x) and the period is divided by 1/x.

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