Discuss the continuity of the following functions at the indicated point(s).

at x = 1

In this problem we need to check continuity at x = a

Given function is

at x = 1 …… Equation 2

∴ we need to check LHL, RHL and value of function at x = a (for idea and meaning of continuity refer to Q10(i) )

Clearly,

f(1) = n–1 [from equation 2]

LHL =

Using binomial theorem –

(1–h)^{n} = 1 – nh + h^{2} – …….

LHL =

Putting h=0 we get,

LHL = n

RHL =

Using binomial expansion as used above we get the following expression

Similarly,

RHL =

Putting h=0 we get,

RHL = n

Thus RHL = LHL ≠ f(1)

∴ f(x) is discontinuous at x=1

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