Continuity and Differentiability 31.12.08 - NCERT

The whole of science is nothing more than a refinementof everyday thinking. ALBERT EINSTEIN IntroductionThis chapter is essentially a continuation of our study ofdifferentiation of functions in Class XI. We had learnt todifferentiate certain functions like polynomial functions andtrigonometric functions. In this chapter, we introduce thevery important concepts of Continuity , Differentiability andrelations between them. We will also learn differentiationof inverse trigonometric functions. Further, we introduce anew class of functions called exponential and logarithmicfunctions. These functions lead to powerful techniques ofdifferentiation. We illustrate certain geometrically obviousconditions through differential calculus.

CONTINUITY AND DIFFERENTIABILITY149 Example 1 Check the continuity of the function f given by f(x) = 2x + 3 at x = 1. Solution First note that the function is defined at the given point x = 1 and its value is 5. Then find the limit of the function at x = 1. Clearly 1 1 lim ( ) lim(2 3) 2(1) 3 5 x x f x x → → = + = + = Thus 1 lim ( ) 5 (1)

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