Question

10. Let f(x)- x+1, when x<0 x21, when x20 Calculate Sro(x) Graph fx) near zero and then graph Sce)(x) near zero. So)(x) fx) lim So (x) = x 0 lim So)(x) x0+ limSt (x) x0 Based on the graph of f(x) and a limit calculation, deternmine if f(x) is continuous at x=0. Based on the graph of So)(x) and the limit calculations above, classify what kind of discontinuity point S(o) (x) has at x-0 Does f '(0) exist? If yes, what does f '(0)-?

Answer 1

The graph of the piecewise function is shown below, both the pieces drawn on separate graphs:

1 -2 2 XtV

Since f(x)= x+1 has x<0, f(x) would not exist at x=0, hence the slope of the secant function ,S₀(x), cannot be drawn, as following calculations would show

This would be the right hand limit (o-() х>ot because X2o In this case x+l-l This would be some positive number very very small close to zero f-Aol This would be the left hand limit x-о would be some number close to 1 but a bit lesser than that Jo) Je)x In this case X As can be seen from the two calculations made above, that left and right hand limits of S,K) exist, but are not equal

This means that lim x->0 S₀(x) does not exist. Function f(x) is therefore not continuous at x=0 and hence not differentiable. f'(0) would therefore not exist. The discontinuity at x=0 is not removable.

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