May 17th, 2017, 10:38 AM  #1 
Senior Member Joined: May 2014 From: Allentown PA USA Posts: 107 Thanks: 6 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(x2) < 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. 
May 19th, 2017, 10:25 AM  #2 
Member Joined: Jan 2016 From: Athens, OH Posts: 89 Thanks: 47 
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 $0<x2<\delta$, it follows that $4/x2<k$. Whoever gave the solution $\delta=2k/(2k)$ 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 $${4\over2+\delta}=2k\text{ for }\delta$$ 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: 
May 22nd, 2017, 01:53 AM  #3 
Senior Member Joined: May 2014 From: Allentown PA USA Posts: 107 Thanks: 6 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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