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.
Finite limit of a function at a real point.
For example, we prove that if
. We look for a
We solve the inequality on the right:
thus, it suffices to choose any
, and we are done.
See Figure fig positive infinite limit of a function at a real point.
Positive infinite limit of a function at a real point.
Negative infinite limit of a function at
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 (
need the same
tools). Without loss of generality, we suppose that
Suppose that the function
tends to both real numbers
is arbitrary close to
. We have:
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