 #1
DumpmeAdrenaline
 35
 0
 Homework Statement:
 Suppose Lim x>c f(x)=0 and that g(x) is bounded by a number K>0 such that g(x)<K. Define h by h(x)=f(x)g(x). Prove that Lim x>c h(x)=0.
 Relevant Equations:

xc<δ > f(x)0<ϵ
g(x)<K
xc<δ > f(x)0<ϵ
g(x)<K
g(x)f(x)0<K*ϵ
The product of the absolute values equals the absolute value of the product.
f(x)g(x))=h(x)<K*ϵ
For a range of input values within δ of c, the corresponding outputs can be made within Kϵ of 0. What if g(x) has a jump at x=c but is bounded by K. If approaching x=c from the right g(x) approaches L1 where L1<K and if approaching x=c from the left g(x) approaches L2 where L2<K. Will the above inequality holds because f(x)=0 and g(x) is bounded though discontinuous fg is again a zero function.
g(x)<K
g(x)f(x)0<K*ϵ
The product of the absolute values equals the absolute value of the product.
f(x)g(x))=h(x)<K*ϵ
For a range of input values within δ of c, the corresponding outputs can be made within Kϵ of 0. What if g(x) has a jump at x=c but is bounded by K. If approaching x=c from the right g(x) approaches L1 where L1<K and if approaching x=c from the left g(x) approaches L2 where L2<K. Will the above inequality holds because f(x)=0 and g(x) is bounded though discontinuous fg is again a zero function.