Can these limits be proven to be equal?

[Bipolarity]

Prove that
\lim_{a→0-}\frac{1}{a} = \lim_{b→0+}\frac{1}{b}

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

 

[Mark44]

Prove that
\lim_{a→0-}\frac{1}{a} = \lim_{b→0+}\frac{1}{b}

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

 

[Bipolarity]

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

 

[Mark44]

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.