Q. 184.0( 2 Votes )

For what value of k is the function continuous at
x = 1?

Answer :

Ideas required to solve the problem:


1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x) , we can say that if we plot the coordinates (x , f(x)) and try to join all those points in the specified region, we can do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve without any breakage.


Mathematically we define the same thing as given below:


A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to be checked


If:–


…… equation 1


where h is a very small positive no (can assume h = 0.00000000001 like this )


It means :–


Limiting value of the left neighbourhood of x = c also called left hand limit LHL must be equal to limiting value of right neighbourhood of x= c called right hand limit RHL and both must be equal to the value of f(x) at x=c f(c).


Thus, it is the necessary condition for a function to be continuous


So, whenever we check continuity we try to check above equality if it holds true, function is continuous else it is discontinuous.


Given,


…… equation 2


We need to find the value of k such that f(x) is continuous at x = 1


Since f(x) is continuous at x = 1


(LHL as x tends to 1) = (RHL as x tends to 1) = f(1)



As, f(1) = k [from equation 2]


We can find either LHL or RHL to equate with f(1)


Let’s find RHL but if you want you can proceed with LHL also.


RHL =


=


As, f(x) is continuous


RHL = f(1)


k = 2


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