HSC Science (General) 12th Board ExamMaharashtra State Board
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# Solution for Discuss the Continuity of the Following Functions. If the Function Have a Removable Discontinuity, Redefine the Function So as to Remove the Discontinuity - HSC Science (General) 12th Board Exam - Mathematics and Statistics

#### Question

Discuss the continuity of the following functions. If the function have a removable discontinuity, redefine the function so as to remove the discontinuity

f(x)=(4^x-e^x)/(6^x-1)  for x ≠ 0

=log(2/3)  for x=0

#### Solution

lim_(x->0^-f(x))=lim_(x->0^-)(4^x-e^x)/(6^x-1)

=lim_(h->0)((4^(0-h)-1)/(0-h)-(e^(0-h)-1)/(0-h))/((6^(0-h)-1)/(0-h))

=(log4-loge)/log6

=log(4/e)/log6

lim_(x->0^+)f(x)=lim_(x->0^+)(4^x-e^x)/(6^x-1)

=lim_(h->0)((4^(0+h)-1)/(0+h)-(e^(0+h)-1)/(0+h))/((6^(0+h)-1)/(0+h))

=(log4-loge)/log6

=log(4/e)/log6

LHL = RHL at x = 0.

But f(0) != lim_(x->0)f(x).

Hence, the given function has removable discontinuity at x = 0.

To remove the discontinuity, we define  f(0)=log(4/e)/log6

So the revised function becomes

f(x)={((4^x-e^x)/(6^x-1), x!=0),(log(4/e)/log6,x=0) :}

Is there an error in this question or solution?

#### APPEARS IN

2015-2016 (March) (with solutions)
Question 5.2.1 | 4.00 marks
Solution Discuss the Continuity of the Following Functions. If the Function Have a Removable Discontinuity, Redefine the Function So as to Remove the Discontinuity Concept: Concept of Continuity.
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