Can these limits be proven to be equal?

  • Thread starter Bipolarity
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  • #1
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Main Question or Discussion Point

Prove that
[tex] \lim_{a→0-}\frac{1}{a} = \lim_{b→0+}\frac{1}{b} [/tex]

Is this statement true? How can one prove its truth/falsity? Would we need to use the precise Cauchy definition of the limit to do this?

BiP
 

Answers and Replies

  • #2
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Prove that
[tex] \lim_{a→0-}\frac{1}{a} = \lim_{b→0+}\frac{1}{b} [/tex]

Is this statement true?
No.
How can one prove its truth/falsity?
A quick sketch of the graph of y = 1/x should convince you that this statement is not true.
Would we need to use the precise Cauchy definition of the limit to do this?

BiP
 
  • #3
775
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No.A quick sketch of the graph of y = 1/x should convince you that this statement is not true.
So I see that from one end it approaches positive infinity, and from the other end it approaches negative infinity.

But is there a way to prove this rigorously without having to refer to a visual aid such as a graph?

BiP
 
  • #4
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5,199
Yes, in the same way that you prove that the limit as x → a of a function is infinity; for each M > 0, there is a ## \delta > 0## such that if |x - a| < ## \delta##, then f(x) > M.

You have to adjust things slightly to deal with left- and right-side limits, and dealing with a limit of negative infinity, but this is the general idea.
 

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