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2. Is the function below continuous at x exist? Defend your answers carefully 2? Does the derivative of f at 2, that is f(2) (r)-は: tos s2

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Answer #1

Answer 2

A function is continuous If LHL = RHL = f(x)

LHL= lim f(x)= lim 22-4

LHL = lim_{x o 2+ }f(x) = lim_{x o 2 }(x^3)/2 = 4

f(2) = 2^2 = 4

Hence LHL = RHL = f(x).

Hence this function is continuous at x = 2

A function is differentiable If LHD = RHD

LHD is given by:

f(x+h)-f(x) lim li lim h-W-

RHD is given by:

f(x+h)-f(x) ((x+h)3 )/2-(2.2)/2 lim h-+0+ = lim = lim 2h 3z2

Hence, LHD = 2x = 4 and RHD = (3x2)/2 = 6

=> LHD is not equal to RHD

=> This function is not differentiable at x = 2. Hence f '(2) does not exist  

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