Limit at one real point.

1. .

See Figure 1.
Let us comment Figure 1(c). The (small) positive number determines the red band; for any value of this , you can find a value of which determines a vertical band, in green, so that the graph of passes through the rectangle determined by the two bands.

For example, we prove that if , then . Take any . We look for a such that

We solve the inequality on the right:

thus, it suffices to choose any such that , and we are done.

2. See Figure fig positive infinite limit of a function at a real point.

Proposition 4.1.2   If has a limit at , this limit is unique.

Proof. This is true, either when the limit is finite or is infinite. We will develop here the proof for the finite case, the reader can do the work for the infinite case ( or need the same tools). Without loss of generality, we suppose that .

Suppose that the function tends to both real numbers and , when is arbitrary close to , i.e.

and

Denote . We have:

Thus:

i.e.

By the triangular inequality, we have:

Hence, the non negative number is less than any positive number, so it must be equal to 0; this means that .

Noah Dana-Picard 2007-12-28