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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 - 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

f(0) = log (2/3)      .....Given .....(1)

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

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

lim_(x->0) ((4^x - 1)/x - (e^x - 1)/x)/((6^x - 1)/x)  .....[x → 0 , x ≠ 0]

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

=((log4) - 1)/log6    .......[because lim_(x->0) (a^x - 1)/x = log e]

therefore lim_(x->0) f(x) = ((log4) - loge)/log6

therefore lim_( x ->0) = [log4]/[loge.log6]

therefore lim_( x ->0) = [log4]/[1.log6]

therefore lim_( x ->0) = log(2/3)

From (1) and (2) , lim_(x->0) f(x) ≠ f(0)

∴ f is discontinuous at x = 0

Here lim_(x->0)  f(x) exists but not equal to f(0). Hence , the discontinuity at x = 0 is removable and can be removed by redefining the function as follows :

f(x)=(4^x-e^x)/(6^x-1)  for x ≠ 0

=log(2/3)  for x=0

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#### 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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