some random calculus

[pHysiX]

Consider the function

1) Give a reason why the quotient rule for differentiation cannot be used to find g'(0)
2) Determine whether or not g is differentiable at x=0 by evaluating separately the left and right hand limits:
and
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3) Why is it that derivatives do not exist at end points?
   *My reasoning is that the limits at the end points do not exist*

Thanks heaps in advance =]

[TrueTears]

2) g(x) is differentiable at x = 0

Consider the original function

when

and when

Sketching these 2 truncas' with the respective domains shows that at x = 0 the graph is continuous.

Now consider the left and right limits

When x is approaching 0 from the negative we are dealing with


When x is approaching 0 from the positive we are dealing with



is differentiable at x = 0

[Mao]

1. |x| is not differentiable at x=0, hence the quotient rule cannot be employed here.

2. as above shown by TT (however, to be pedantic, you should also show g to be continuous at x=0)

3. cannot draw a tangent through an endpoint (one of the left hand or right hand limit does not exist, hence not differentiable)