The limit of a trigonometric function.

In summary, sin(1/x) does not have a limit as x approaches 0 because it oscillates between -1 and 1 and the limit does not exist because there are two values.
  • #1
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Homework Statement


Find the limit of sin(1/x) as x -> 0, I don't really understand why there isn't a limit, wouldn't it be +/- 1? or it doesn't exist because there are two values? I just need a more thorough explanation of the answer really.
 
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  • #2
Essentially, yes the limit doesn't exist because there are two values. More specifically, the sine function is oscillating as you extend towards infinity; not just between its two extreme values (1 and -1), but everything in between as well.
 
  • #3
icesalmon said:

Homework Statement


Find the limit of sin(1/x) as x -> 0, I don't really understand why there isn't a limit, wouldn't it be +/- 1? or it doesn't exist because there are two values? I just need a more thorough explanation of the answer really.

If x is approaching 0, 1/x is approaching infinity or -infinity, depending if x is positive or negative. What does sin do if the value goes to infinity? -infinity?
 
  • #4
no it approaches 0... I see now that the reciprocal makes it larger while the values approaching infinity make it signifigantly smaller. But wait, aren't real numbers paired with circular functions interpreted as radians if pi isn't present? Wouldn't this just mean the value of x represents a reference arc of 360 degrees or 2pi? How does that value go off unto infinity when it's domain is locked in at [-1,1] such that it's amplitude is equal to one. I'm sorry if how this is in some way phrased unclearly.
 
  • #5
icesalmon said:
no it approaches 0... I see now that the reciprocal makes it larger while the values approaching infinity make it signifigantly smaller. But wait, aren't real numbers paired with circular functions interpreted as radians if pi isn't present? Wouldn't this just mean the value of x represents a reference arc of 360 degrees or 2pi? How does that value go off unto infinity when it's domain is locked in at [-1,1] such that it's amplitude is equal to one. I'm sorry if how this is in some way phrased unclearly.

You're looking at this way too hard. Let's look at a similar example. If you have sin x as x-> infinity, we get oscillation between -1 and 1 and this occurs forever so the limit does not exist. So now looking at sin 1/x as x-> 0, 1/x goes can go infinity or negative infinity, so...
 
  • #6
I see, the infinite behavior arises from the fact that these are waves. Now let me ask, how would this change if the function was tangent or cotangent? the domain isn't continuous. Where there are asymptotes at every npi or npi/2 value. How could the limit be interpreted if the range was infinity?
 
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  • #7
icesalmon said:
I see, the infinite behavior arises from the fact that these are waves. Now let me ask, how would this change if the function was tangent or cotangent? the domain isn't continuous. Where there are asymptotes at every n or n/2 value of pi. How could the limit be interpreted if the range was infinity?

If you can, look at the graph of tangent. From -pi to pi, the y values range from -infinity to +infinity. From pi to 3pi, the y values range from -infinity to +infinity, ...

Also notice that (every odd multiple of pi)/2 is a vertical asymptote

So if x approaches infinity, there's no way we're going to get an actual limit for tangent.
 
  • #8
yeah it makes more sense now, interpreting the limits of circular functions, thank you for your time!
 

1. What is the limit of a trigonometric function?

The limit of a trigonometric function is the value that a function approaches as the input (x) approaches a certain value. It can also be thought of as the value that the function "approaches" as the input gets closer and closer to a specific value.

2. How is the limit of a trigonometric function calculated?

The limit of a trigonometric function can be calculated using various methods such as substitution, factoring, and trigonometric identities. These methods involve manipulating the function algebraically to simplify it and then evaluating the function at the given value.

3. What is the significance of the limit of a trigonometric function?

The limit of a trigonometric function is significant as it helps us understand the behavior of the function near a certain point. It can also be used to determine if the function is continuous at that point or if there is a discontinuity.

4. Can the limit of a trigonometric function be undefined?

Yes, the limit of a trigonometric function can be undefined. This can occur if the function has a vertical asymptote or if the limit approaches different values from the left and right sides of the given point.

5. How can the limit of a trigonometric function help in solving real-world problems?

The limit of a trigonometric function can be used in solving real-world problems involving rates of change and growth. For example, in physics, the limit of a function can be used to determine the velocity of an object at a specific point in time.

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