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flying2000
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How to prove there is no limit(x->0) of cos1/x using theorm of limit?
Anybody can give me some hints?
thanks
Anybody can give me some hints?
thanks
Last edited:
Dr-NiKoN said:I'm just learning this myself, but here is how I would approach it.
[itex]\lim_{x\to 0}f(x)=\cos(\frac{1}{x})[/itex]
You know what happens to [itex]\lim_{x\to 0}f(x) = \frac{1}{x}[/itex]
You also know what a graf with [itex]f(x) = \cos(x)[/itex] looks like.
Now consider
[itex]y = \frac{1}{x}[/itex]
and
[itex]\lim_{y\to\infty}f(y)=\cos(y)[/itex]
matt grime said:Use the easier, and equivalent, definition of limit.
If you can find two sequences a(n) and b(n) tending to zero such that cos(a(n)) and cos(b(n)) tend to different numbers you're done.
matt grime said:To prove the counter example it suffices to show that given any d there is a e such that there is some x with |cos(x)-L| > e and |x|<d
but this is trivial. firstly L must be between -1 and 1, and let x be some sufficiently large solution to cos(x)=1 or -1, then one of |1-L| and |-1-L| must be greater than 1 (which we can choose to be e)
nb e:=epsilon, d:=delta
The theorem of limit, also known as the limit theorem, is a fundamental concept in calculus that states if a function f(x) approaches a specific value L as x approaches a certain value c, then we can say that the limit of f(x) as x approaches c is equal to L. This is denoted as "lim f(x) = L" or "f(x) → L" as x → c.
The theorem of limit can be used to determine the limit of any function, including the cosine function, as long as the function satisfies the conditions for the theorem to be applied. In the case of cos1/x, the limit can be evaluated as x approaches 0, since this is the only point where the function does not exist.
Yes, the limit of cos1/x can be proven using the theorem of limit. By applying the definition of the limit, we can show that the limit of cos1/x as x approaches 0 is equal to 1, which means that there is no limit as x approaches 0.
In order for the theorem of limit to be applied, the function must be defined and continuous in the interval around the limit point, except possibly at the limit point itself. Additionally, the limit point must be a real number or infinity.
To evaluate the limit of cos1/x using the theorem of limit, we first rewrite the function as cos(1/x) and then apply the limit definition. This involves finding the limit of cos(1/x) as x approaches 0 from both the left and right sides. If the limits from both sides are equal, then the overall limit exists and is equal to that value.