Real Analysis Answers

suppose f and g are differentiable at a. Then, prove f/g is differentiable at a, and
(f/g)'(a)= f'(a)g(a)-f(a)g'(a)/ (g(a))^2.

x belongs to r which satisfy the inequality |x|+|x+1|<2 find all x

Interpret and prove:
(i) o+o=o
(ii) o+O=O
(iii) O+O=O
(iv) o(O)=o[(iv) and (v) represent compositions of functions]
(v) o x O=o

Show that f=O(1) [this is the letter O not the number 0] as x->a if and only if f(x) is bounded on some neighborhood of a.

Show that f =o(1) as x->a if and only if f(x)->0 as x->a.

The statement "If f = o(g) then f = O(g)" can be abbreviated o⇒O. Prove this.

Show that K(x-a) = O(k-a) [this is the letter O no the number 0] for any K.

If K is a constant, show that K(x-a) = o(x-a) [yes that is a "o" not a "0"] if and only if K=0.

Task If (a sub_n) and (b sub_n) are cauchy sequences, show directly that (a sub_n + b sub_n) is also a cauchy sequence