Limits when there is a sine function?

In summary, the conversation discusses finding the limit of x*sin(1/x) as x approaches infinity. The suggested method is to use a substitution and then apply a well-known limit to solve. L'Hopital's Rule is not necessary for this problem.
  • #1
applestrudle
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Homework Statement



lim x-> ∞ xsin(1/x)

Homework Equations


The Attempt at a Solution



I know that this is an ∞.0 type limit but I can't figure out how to change the sin function.

Thank you
 
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  • #2
Hint
[tex]x \sin(1/x)=\frac{\sin(1/x)}{1/x}.[/tex]
 
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  • #3
vanhees71 said:
Hint
[tex]x \sin(1/x)=\frac{\sin(1/x)}{1/x}.[/tex]

Ahh thank you! Then you use L'Hopitals Rule?
 
  • #4
I wouldn't. This limit is related to this well-known limit
$$\lim_{t \to 0}\frac{sin(t)}{t}$$
 
  • #5
Mark44 said:
I wouldn't. This limit is related to this well-known limit
$$\lim_{t \to 0}\frac{sin(t)}{t}$$

What do you mean?
 
  • #6
Hint 2: Use a substitution, then figure out what the appropriate change in the limit would be.
 

1. What is a limit when there is a sine function?

A limit when there is a sine function is the value that a function approaches as the input approaches a specific value. In other words, it is the value that the function "approaches" as we get closer and closer to a certain point on the graph.

2. How do you find the limit of a sine function?

To find the limit of a sine function, you can use the following steps:
1. Substitute the value that the input is approaching into the function.
2. Simplify the resulting expression.
3. If the resulting expression is undefined, use algebraic techniques to simplify it.
4. If the resulting expression still cannot be evaluated, the limit does not exist. Otherwise, the value of the expression is the limit.

3. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the values of the function as the input approaches from one direction (either from the left or right) of the specific value. A two-sided limit, on the other hand, considers the values of the function as the input approaches from both directions of the specific value.

4. Can a sine function have a limit at a discontinuity?

No, a sine function cannot have a limit at a discontinuity. A discontinuity occurs when the function has a break or a hole in its graph, and in this case, the function does not approach a specific value as the input approaches the discontinuity.

5. How do limits of sine functions relate to the unit circle?

Limits of sine functions relate to the unit circle because the values of sine are the y-coordinates of points on the unit circle. The limit of a sine function at a specific value can be determined by looking at the y-coordinate of the point on the unit circle that corresponds to the input value.

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