MyMaths

                                                        CONVERGENCE  

                                                                                      CONTINUITY IF Lim f(x)= Lim  f(x) =  f(a)     x->a-         x->a+ THEN    FUNCTION IS CONTINUOUS AT 'a' WHERE  ,    Lim  f(x) is a left hand limit                       x->a-                       Lim f(x) is a right hand limit                       x->a+                           REMOVABLE DISCONTINUITY IF Left hand limit = Right hand limit But   f(a) is different; then...

                               L'HOSPITAL'S                                                                 RULE IF  Lim  F(X)/G(X) = 0/0 ;       X->a THEN   Lim F(X)/G(X)  =  Lim F'(X)/G'(X)                X->a                         X->a PROBLEMS BASED ON ABOVE RULE:                                   _ 1. Lim   sin(1-x)/√x - 1 = ?    x->1 ANSWER: IF WE PUT x=1; WE GET  ...........  sin(1-x)                         ...

                                                                 DIFFERENTIATION  THE TRIANGLE FORMED ON TOP IS               tanθ= dy/dx..............from above figure              as slope is tanθ is slope(say m);              so  m= dy/dx.              as dy = F(X+dx) - F(X);      FROM FIRST FIGURE.              SO     m= F(X+dx)-F(X)/dx         ...

CIRCLE DERIVATION OF AREA OF CIRCLE LETS TAKE d𝞡 , A  VERY SMALL ANGLE IN ABOVE FIGURE SO; THE SMALL PART OF CIRCLE IN DIAGRAM  CAN BE CONSIDERED AS RIGHT ANGLED TRIANGLE FROM ABOVE FIGURE, tan d𝞡= opposite side/ adjacent side                                                     tan d𝞡   = ds/r AS ANGLE IS VERY SMALL, tan d𝞡 = d𝞡  SO,                                               d𝞡 =ds/r SO,                                                ds=r d𝞡 ..................(1)                                     ...