Q. 54.6( 5 Votes )

If Find whether f(x) is continuous at x = 0.

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 the 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 i.e. 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, function is continuous else it is discontinuous.


2. The idea of sandwich theorem – This theorem also known as squeeze theorem that you may have encountered in your class 11 in limits chapter suggests that


If I be an interval having a point as a limit point. Let g, f, and h be functions defined on I, except possibly at a itself. Suppose that for every x in I not equal to a, we have{\displaystyle \lim _{x a}g(x)=\lim _{x a}h(x)=L.}{\displaystyle \lim _{x a}f(x)=L.}


g(x) ≤ f(x) ≤ h(x) and also



Then , We say that f(x) is squeezed between g(x) and h(x) or you can assume it like sandwich.



equation 2


NOTE : denominator in the above limit should be exactly same as that of content in sine function


Eg :


Let’s solve :


To check whether function is continuous at x=0 we need to check whether LHL = RHL = f(c)


As continuity is to be checked at x = 0 therefore c=0.


As, Equation 3


Clearly, f(0) = 1 using eqn 3


LHL =


Using equation 3 –


[ sin(–θ) = – (sin θ)]


To find its limit we need to think of Sandwich theorem as the form looks similar but term in denominator is not exactly same as that in the content of sine function.


Ok no problem lets bring it there but if we put 3 in denominator we need to put a 3 in numerator too so that both can be cancelled.


Let’s do it:


LHL


RHL =


Using equation 3 –



Again using sandwich theorem as we used while finding LHL


RHL


Thus,


LHL = RHL = 3


But, f(0) = 1 by definition of f(x) i.e. equation 3


LHL = RHL ≠ f(0)


condition for function to be continuous is not satisfied


f(x) is discontinuous at x = 0


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