# Can these limits be proven to be equal?

## Main Question or Discussion Point

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
Mentor
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

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
Mentor
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.