Example 4.20 Let

f:(1,\\\\infty )->R

be defined by

f(x)=(1)/(x-1)

. If\

x_(n)=1+(1)/(n)

for each

n

in

N

, then

x_(n)->1

, but

f(x_(n))=n

for each

n

. Hence,\

(f(x_(n)))_(ninN)

is not Cauchy. By Theorem 4.5,

f

is not uniformly continuous on\

(1,\\\\infty )

. [This sequence approach always works when the function has a vertical\ asymptote. The reader should redo Example 4.18 using this approach.]

Example 4.20 Let f:(1,)R be defined by f(x)=1/(x1). If xn=1+1 for each n in N, then xn1, but f(xn)=n for each n. Hence, (f(xn))nN is not Cauchy. By Theorem 4.5, f is not uniformly continuous on (1,). [This sequence approach always works when the function has a vertical asymptote. The reader should redo Example 4.18 using this approach.]