Apply the Limit Theorems in "Intro to Real Analysis"

2. Let f : (0, co) -+ R and a > 0 be a fixed number. Suppose lim,-.. aa f (a) = L for some L E R. Show that limx- f (x) = 0.9. Show that if f : (a, co) - R is such that lim xf(x) = L where L E R, then lim f(x) = 0.Step 1/3 Let f : (a,0) -, be a function such that *- lim xf (x) = L, where LED Then we see that for any &>0, there exist K > 0, such that for any * (A,09), we have |xf ( x) - L 0 there exists K = K(&) > a such that for any x > K, then If ( x) - L|