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 = 1

Given function is

at x = 1 …… Equation 2

∴ we need to check LHL, RHL and value of function at x = 1 (for approaching idea and meaning of continuity refer to Q10(i))

Clearly,

f(1) = 2 [ from equation 2]

LHL =

Since h is positive no which is very close to 0

∴ (h–2) is negative and hence h(h–2) is also negative.

∴ |h(h–2)| = –h(h–2)

∴LHL =

RHL =

Since h is a positive no which is very close to 0

∴ (h+2) is positive and hence h(h–2) is also positive.

∴ |h(h+2)| = h(h+2)

∴ RHL =

Clearly, LHL ≠ RHL

∴ f(x) is discontinuous at x=1

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