HSC Arts 12th Board ExamMaharashtra State Board
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Discuss the Continuity of the Following Functions. If the Function Have a Removable Discontinuity, Redefine the Function So as to Remove the Discontinuity - HSC Arts 12th Board Exam - Mathematics and Statistics

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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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