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 January 5th, 2019, 06:26 AM #1 Member   Joined: Apr 2017 From: India Posts: 39 Thanks: 0 limit Using delta- epsilon definition, please give me a proof of how limit x tends to infinity of the function f(x) = 1/x is not equal to 1. I am confused in the choice of delta.
 January 5th, 2019, 02:14 PM #2 Global Moderator   Joined: May 2007 Posts: 6,663 Thanks: 649 You can do the mechanics. $\lim_{x\to \infty} |\frac{1}{x}-1|=1\ne 0$. Thanks from topsquark and idontknow
January 5th, 2019, 06:33 PM   #3
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The definition of the limit says that $$\lim_{x \to \infty} \frac1x = 1$$
means that
Quote:
 for every $\epsilon > 0$ there exists an $x_0$ such that $\left|\frac1x - 1\right| < \epsilon$ for every $x > x_0$.
You have to prove that statement to be false. This entails finding a value of $\epsilon$ for which no such $x_0$ exists - any $x_0$ you pick has some value of $x > x_0$ such that $\left|\frac1x - 1\right| \not < \epsilon$.

I suggest that you consider some value of $\epsilon < 1$ - for example $\epsilon = \frac12$.

Last edited by v8archie; January 5th, 2019 at 06:41 PM.

 January 6th, 2019, 07:38 AM #4 Senior Member   Joined: May 2016 From: USA Posts: 1,252 Thanks: 519 Alternatively, you could prove that the limit is something other than 1. Thanks from topsquark
January 6th, 2019, 07:42 AM   #5
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Quote:
 Originally Posted by JeffM1 Alternatively, you could prove that the limit is something other than 1.
You'd still need to prove that the limit is unique.

January 6th, 2019, 10:10 AM   #6
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Quote:
 Originally Posted by Micrm@ss You'd still need to prove that the limit is unique.
Ahh, yes.

January 6th, 2019, 05:06 PM   #7
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Quote:
 Originally Posted by Micrm@ss You'd still need to prove that the limit is unique.
If there is a limit, it must be unique(??)

January 6th, 2019, 08:26 PM   #8
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Quote:
 Originally Posted by Jomo If there is a limit, it must be unique(??)
Sure, but that needs to be proven.

January 7th, 2019, 11:42 AM   #9
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Quote:
 Originally Posted by Micrm@ss Sure, but that needs to be proven.
Of course, it could be proved generally as a theorem and then applied to specific problems, but I must admit that I merely assumed the general theorem and neglected to prove it.

 January 7th, 2019, 11:57 AM #10 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,900 Thanks: 1094 Math Focus: Elementary mathematics and beyond The derivative of f(x) = 1/x, -1/x$^2$, tells us f(x) is monotone decreasing. Is that sufficient to tell us the limit is unique (if it exits)? I think it is.

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