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Rob S.

asked • 07/28/18

Limit of a cotangent function as x approaches infinity?

Is the limit well defined?
Why or why not?

the limit as x → ∞ of Cot ((x^2+1)/(x+3))

1 Expert Answer

By:

Paul M.

tutor
I did give this question some more thought.
If f(x) = [(x2 + 1)/(x + 3)] and g(x) = x,
the lim as x→∞ of f(x)/g(x) = 1 (and I think I could get an ε X  argument to prove that...it would, I think, be messy).
 
What this says is that even though f(x) does NOT approach a limit, the ratio does.  The problem with situations like this one is that even though the ratio approaches 1, the absolute difference may be quite large, that is |f(x) - g(x)| may be large.
 
The same argument will apply to cot[f(x)]/cot x...this ratio will also have limit 1 although I would be hard pressed to prove that.  Nevertheless, when you look at the graph of cot[f(x)]/cot x, you will see that cot{f(x)] looks like cot x visually.
 
I hope this helps!
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07/30/18

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