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May 17th, 2017, 10:38 AM   #1
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Math Focus: dynamical systen theory
problem with a calculus exercise

Dear MyMathForum Community:

I tried understanding this problem, but to no avail.
Find an appropriate number L, and a deleted neighborhood N of a, such
that given k > 0,

L - k < f(x) < L + k

here, f(x) = 4/x, and a = 2. The correct answer to this exercise is:
0 < abs(x-2) < 2k/(2 - k). I cannot understand how the author got this answer. Thank you.


Best regards,
Carl Mesaros

Last edited by skipjack; May 18th, 2017 at 05:06 AM.
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May 19th, 2017, 10:25 AM   #2
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You didn't state the problem very well. Since this is calculus, I assume you mean: for a=2, L=2 and for any k>0, find $\delta>0$ such that for any x with $0<|x-2|<\delta$, it follows that $|4/x-2|<k$.

Whoever gave the solution $\delta=2k/(2-k)$ is incompetent (I hope not the author of the text). First if $k>2$, $\delta$ isn't even positive. Here's how the incorrect $\delta$ was found: Solve $${4\over2+\delta}=2-k\text{ for }\delta$$

For this particular function, you can solve
$${4\over 2-\delta_1}=2+k\text{ for }\delta_1$$
Then a correct deleted neighborhood is $(2-\delta_1,2+\delta)\setminus \{2\}$
The only reason this works is that the function is decreasing, so it is not a general good procedure.

Here's a graph that shows that $\delta$ is wrong:

Thanks from jonah and dthiaw
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May 22nd, 2017, 01:53 AM   #3
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From: Allentown PA USA

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Math Focus: dynamical systen theory
An apology

Dear Johng40:
You are right. I did not state the problem correctly. My apologies I offer
to you. The basic problem I had is how to manipulate 4/x in a way that would
make the solution of the problem possible. Thank you.


Best regards,
Carl Mesaros
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