- #1
pivoxa15
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Homework Statement
lim(x->infinity) sin(x)
The Attempt at a Solution
undefined.
Can the Limit of Sin(x) as x Approaches Infinity be Proven Graphically?
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In summary, the conversation discusses the limit of the function sin(x) as x approaches infinity. The participants argue that the limit is undefined and cannot be determined without further information. They suggest using the sequential criterion for limits to determine the existence of the limit, and point out that the function has a finite period and therefore will oscillate between its range, making the limit non-existent. They also mention the importance of looking at the definition of the function and considering its behavior near infinity.
lim(x->infinity) sin(x)
undefined.
- #2
CompuChip
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Why do you think that?
Please give an argument.
And please say what knowledge you have of limits (basis calculus, real analysis, etc)
Please give an argument.
And please say what knowledge you have of limits (basis calculus, real analysis, etc)
- #3
ZioX
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Can you give a proof why this limit doesn't exist?
The sequential criterion for limits is handy, if you know this criterion.
The sequential criterion for limits is:
[itex]\lim_{x \to a}f(x)=b[/itex] iff for every sequence x_n in the extended real numbers (abstractly, the domain of f) converging to a has the property that f(x_n) converges to b.
Can you think of a sequence converging to infinity such that sin(x) converges? Can you think of another sequence converging to infinity such that sin(x) converges to a different value?
The sequential criterion for limits is handy, if you know this criterion.
The sequential criterion for limits is:
[itex]\lim_{x \to a}f(x)=b[/itex] iff for every sequence x_n in the extended real numbers (abstractly, the domain of f) converging to a has the property that f(x_n) converges to b.
Can you think of a sequence converging to infinity such that sin(x) converges? Can you think of another sequence converging to infinity such that sin(x) converges to a different value?
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- #4
Sartre
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I don't think you should just say that the limit is something or other. If you want to take limits in simple non-series functions, then you should look at the outer rims of the function.
For example sin(x) is defined as -1 < x < 1. Now you must work out if the limit is negative or positive infinity. And you should look at the definition that ZioX posted. It comes in handy.
For example sin(x) is defined as -1 < x < 1. Now you must work out if the limit is negative or positive infinity. And you should look at the definition that ZioX posted. It comes in handy.
- #5
CompuChip
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Or doesn't existSartre said:
Which is what he said (just didn't prove yet).
- #6
HallsofIvy
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In other words, everyone is saying "yes, the limit doesn't exist, but you must tell why it doesn't exist"!
- #7
Sartre
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CompuChip said:Or doesn't exist
Which is what he said (just didn't prove yet).
Actually I reviewed this part of calculus a couple of days ago. So it is my bad om that one ;)
And the simplest proof of seeing that the limit doesn't exist is graphical.
- #8
uiulic
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Ziox showed it doesn't exist.
x=2n*(pi)
x=(2n+1/2)*(pi)
when n->infinite, then got two different values.
x=2n*(pi)
x=(2n+1/2)*(pi)
when n->infinite, then got two different values.
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- #9
HallsofIvy
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This is non-sense. x can be any real number. I think what you meant was that if y= sin(x) then [itex]-1\le y\le 1[/itex]. And you don't need to "work out if the limit is negative or positive infinity"- it's neither one, it just doesnt' exist.Sartre said:
- #10
Gib Z
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The point is that the limit as x approaches infinity of finite period functions does not exist, as it will oscillate between the range of the function.
- #11
CompuChip
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Sartre said:
Depends what you call a proof. And how rigorous a proof you want.
FAQ: Can the Limit of Sin(x) as x Approaches Infinity be Proven Graphically?
What is a limit in mathematics?
A limit in mathematics is a fundamental concept used to describe the behavior of a function as its input approaches a specific value. It is denoted by the symbol "lim" and is used to determine the value that a function approaches as its input value gets closer and closer to a particular point.
How do you evaluate a limit?
To evaluate a limit, you need to first determine the behavior of the function as the input approaches the given value. You can do this by plugging in values that are very close to the given value and observe the corresponding output. If the function approaches a specific value, then that value is the limit. If the function approaches different values from the left and right sides, then the limit does not exist.
What is the purpose of evaluating a limit?
The purpose of evaluating a limit is to understand the behavior of a function at a specific point. This can help in determining the continuity of a function, finding the slope of a tangent line, and determining the convergence of a series, among other applications.
What are the common techniques for evaluating limits?
There are several techniques for evaluating limits, including direct substitution, factoring, rationalization, and using trigonometric identities. Other methods include using L'Hopital's rule, the squeeze theorem, and the limit definition of a derivative.
Can all limits be evaluated using algebraic techniques?
No, not all limits can be evaluated using algebraic techniques. Some limits, such as limits involving trigonometric or exponential functions, may require the use of advanced techniques or numerical methods to evaluate them. In some cases, the limit may not exist or may be undefined.