Give example of functions f and g such that lim_(xrarrc)[f(x)*g(x)] exists but either lim_(xrarrc)f(x) or lim_(xrarrc)g(x) does not?

2 Answers
Mar 19, 2018

One example is f(x)=1/x and g(x)=arctan(1/x)

lim_(xto0)1/x="undefined" (since it goes to +-oo, oo from the right and -oo from the left)
lim_(xto0)arctan(1/x)="undefined" (since it goes to +-pi/2, pi/2 from the right and -pi/2 from the left)

lim_(xto0)arctan(1/x)/x=+oo (since ooxxpi/2=oo and -pi/2xx-oo=oo)

Mar 19, 2018

c=0 and f(x) = x and g(x) = 1/x