limit function

Question

Let

f(x)=1. x^2sin(1/x^2)+bcosx if x>0

1. 0 if x=0
2. |x|/x+a(x/sqrtx^2) if x<0

evaluate lim x->0- f(x) and write lim x->0+ f(x) in terms of a and b;

if lim x->0 f(x) exists, find the relationship of a and b;

if lim x->0 f(x) =0,find the value of and b

Answer 1

1. Lim x→0^- uses |x|/x+a(x/√x²), which becomes -1+a or a-1; lim x→0⁺ uses x²sin(1/x²)+bcos(x), which becomes 1+b.
2. If lim x→0 exists, then lim x→0^-=lim x→0⁺, so a-1=1+b, a-b=2 is the relationship between a and b.
3. If, as lim x→0, f(x)=0 then, a-1=0 and 1+b=0, so a=1 and b=-1, also satisfying a-b=1-(-1)=2.