Question

I am struggling with this,


Prove the theorem that if limx→d h(x) = P and limx→d k(x) = Q and h(x) ≥ k(x) for all x in an open interval containing d, then P ≥ Q by using the formal definition of the limit, showing all work.

Answer 1

Suppose Q<P, let
\[
a= \frac {P-Q}3
\]
We can find an x close to d so that
\[
|h(x)-P| < \frac a 3 \\
|k(x)-Q| < \frac a 3 \\
\] This imples
that
\[
k(x) < h(x)
\]
A contradiction which implies what we want.

Answer 2

Thanks!!

Answer 3

YW