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Question

Let f(x) be a function defined by fx=3xx+2x,x0 0,x=0.
Show that limx0 fx does not exist.

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Solution

fx=3xx+2x,x0 0,x=0Left hand limit:limx0- 3xx+2xLet x=0-h, where h0.limh0 3-h-h+2-h=limh0 -3hh-2h=limh0 -3h-h=3Right hand limit:limx0+ 3xx+2xLet x=0+h, where h0.limh0 3hh+2h=limh0 3hh+2h=1 limx0- 3xx+2xlimx0+ 3xx+2xThus, limx0 fx does not exist.

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