Differential Calculus by Narayan S.

By Narayan S.

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311. Continuity of a function in an interval. In the last articles, at we dealt with the definition of continuity of a function /(x) now extend this definition its interval of definition. We a point of to continuity in an interval and say that A function /(x) is continuous in an interval [a, &], if it is continuous at every point thereof. Discontinuity. A function /(x) which said to be discontinuous for is not continuous for x=c is xc. The notion of continuity and discontinuity ted by means of some simple examples.

V) log [ 2/i-f-l *) (x-H)/(x-l) (v/0 (sin x)*. 2 (ix) log [x+>J(x ~ 1) n being any integer. ; ] ]. Find out the intervals in 3. cally increasing or decreasing +*)/(! -x) (iv) log (v/) log (tan-'x [ (1 (viii) (Iog 6 x)^. (x) x tan-ix. ] . which the following functions are monotoni- : (02*. 4. (//) (i)*- (ill) 3 1 ''*' (iv) Distinguish between the two functions 1 x, G) . 4 l/x2 ' (v) 2sinx. CHAPTER III CONTINUITY AND LIMIT Introduction. The statement that a function of x is defined in a certain interval means that to each value of x belonging to the interval there corresvalue ponds a value of the function.

Since a*~\\a~* we see that a* is positively very small when x is negatively very large and can be made as near zero as we like provided we give to x a negative ', . value which is sufficiently large nu- merically. With the help of these facts, we can draw the graph which is as shown in (Fig. 19). Fig. 19. 2-42. (i) LetQ

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