Pure Mathematics 2Q

Continuity and Differentiability

1.Let f(x) be a function defined by

f(x)=(1-cosx)/x^2  (if x does not equal to zero)
and
f(x)=1/2                  ( if x equal to zero)

Prove that f(x)is continuous at x=0                     (我唔知點解化簡唔到= =)

2.Let f(x) be a function defined by

f(x)=(X^2-a^2)/a ,(if 0<x<a and x=a)
and
f(x)=a(x^2-a^2)/x^2  ,(if x>a)

(a)show that f(x)is continuous at x=a and sketch the graph os f.

show that f(x) is differentiable at x=a

(c)find f ' (x)for x>0

(d)Is f ' (x)continuous at x=a?

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