[7] Problem 3 Definition: A function f : IR - R is said to vanish at infinity if lim f(x) = lim f(x) =0. I-+ 400 I )-00 (a) Prove that if f : R - R is a continuous function that vanishes at infinity, then f is uniformly continuous on R. (b) Is the conclusion in (a) still valid if lim f(x) = L and lim f(r) = L_are non-zero I-) +00 I -00 real numbers? Give a brief justification or counterexample.Am 3 ) 6 ) Given lim f (X ) = L , and lim B ( x ) = L. x -+ Q x -)-d where Ly and L_ are non-zero real numbers and f is continious . Now Let 8 70 lim 8 ( 2) = L+ x -) + of so I M. SO such that 1 8 ( x ) - L + J K -2 V X > M Now for x, y > M we hove 18 (x ) - 8 (4 ) 1 = 16 (x) - L x - 6(x) + L+/ 5 16 ( x ) - 2 + / + 18(2) - 2+1