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))/x-0 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
lim-x = 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)]/[x-0] = lim xsin(1/x)/x = lim sin(1/x) DNE from b)

e. lim [h(x)-h(0)]/ [x-0] = 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