
Past Exam Paper help!!
Ok
the functions f(x) and g(x) are defined by the formulae
f(x) = sin 1/x g(x) = x sin 1/x for x does not equal 0
f(0) = g(0) = 0
(b) Give the values of limx>0 f(x) and limx>0 g(x), if they exist, with brief reasons for your answers.
(c) Which of the functions f(x) and g(x) is continuous at x = 0?
(d) use the difference quotient (g(x)  g(0))/x0 to show that g(x) is not differentiable at x = 0
(e) Use a difference quotient to show that the function h(x) defined by h(x) = x^2 sin 1/x for x (does not equal) 0 and h(0) = 0 is differentiable at x = 0
(f) Use the substitution u = 1/x to evaluate limx>[infinity] g(x)
:)

b.
 x < xsin(1/x) < x
as x>0
limx = limx = 0
therefore lim xsin(1/x) = 0
limx>0 (sin(1/x) = lim x>inf sin(x) DNE
c. g(x)
d. lim [g(x)g(0)]/[x0] = lim xsin(1/x)/x = lim sin(1/x) DNE from b)
e. lim [h(x)h(0)]/ [x0] = lim x^2sin(1/x) /x = lim xsin(1/x) = 0 from b)
f.limx>inf [g(x)] = lim x>0 [sin(x)/x] = 1