- #1

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[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

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- Thread starter Bipolarity
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- #1

- 775

- 1

[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

- #2

Mark44

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No.Prove that

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

Is this statement true?

A quick sketch of the graph of y = 1/x should convince you that this statement is not true.How can one prove its truth/falsity?

Would we need to use the precise Cauchy definition of the limit to do this?

BiP

- #3

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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

Mark44

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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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