If for find the value which can be assigned to f(x) at x = π/4 so that the function f(x) becomes continuous every where in [0, π/2].


Basic Concept:


A real function f is said to be continuous at x = c, where c is any point in the domain of f if :


where h is a very small ‘+ve’ no.


i.e. left hand limit as x c (LHL) = right hand limit as x c (RHL) = value of function at x = c.


This is very precise, using our fundamental idea of limit from class 11 we can summarise it as, A function is continuous at x = c if :



Function is defined for [0,π] and we need to find the value of f(x) so that it is continuous everywhere in its domain (domain = set of numbers for which f is defined)


As we have expression for x ≠ π/4, which is continuous everywhere in [0,π],so


If we make it continuous at x = π/4 it is continuous everywhere in its domain.


Given,


……….equation 1


Let f(x) is continuous for x = π/4




= = [ tan (π/2–θ) = cot θ ]


= [multiplying and dividing by π/4–x and π/2–2x to apply sandwich theorem]


=


[


value that can be assigned to f(x) at x = π/4 is


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