Question

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?

Answer 1

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`)

Answer 2

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