Find an example of a pair of functions f and g and a point "a"......?

Question

satisfying f(x) < g(x) for all x not equal to a but lim as x-->a of f(x) = lim as x-->a of g(x).

Please show how you got it and whether the pair of functions work or not.
Thanks for your help =)

Answer 1

f(x) = -x^2
g(x) = x^2
a = 0

since f(x) < 0 for x != 0 and g(x) > 0 for x != 0, then clearly f(x) < 0 < g(x) for x != 0.

Since f and g are continuous, lim x->0 f(x) = lim x->0 g(x) = 0

Answer 2

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

g(x) = (x)/(x+1)

point "a" = -1

Use l'Hopital's rule to evaluate the limit as x --> -1, and the limits of these two functions will be equal

Hope this helps ;)