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

at x = 0

In this problem we need to check continuity at x = 0

Given function is

at x = 0

∴ we need to check LHL, RHL and value of function at x = 0 (for idea and meaning of continuity refer to Q10(i))

1. NOTE : Idea of logarithmic limit and exponential limit –

……equation 1

…… equation 2

You must have read such limits in class 11. You can verify these by expanding log(1+x) and e^{x} in its taylor form.

Numerator and denominator conditions also hold for this limit like sandwich theorem.

E.g :

But ,

Now we are ready to solve the problem.

Given function is

at x = 0 …… Equation 3

Clearly,

f(0) = 7 [from equation 2]

LHL = [ putting x = –h in equation 3]

=

Using logarithmic and exponential limit as explained above, we have:

LHL =

RHL = [ putting x = h in equation 3]

=

Using logarithmic and exponential limit as explained above, we have:

RHL =

Thus, LHL = RHL ≠ f(0)

∴ f(x) is discontinuous at x = 0

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