RD Sharma - Mathematics (Volume 1)

Book: RD Sharma - Mathematics (Volume 1)

Chapter: 9. Continuity

Subject: Maths - Class 12th

Q. No. 10 of Exercise 9.1

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10

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 ex 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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