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CONTINUITY, LHD -> Left Hand Derivative, Algebra of Infinity, Lt, f (a - h) - f(a), (a), n + 00, 00, LHD, h -→ 0, - h, (b), %3D, n- 00, RHD - Right Hand Derivative, (c), n x 0, Lt, f (a + h) - f (a), RHD, %3D, 0 (n> 1), 00, (d), h ->0, (e), %3D, If, LHD = RHD, Then f(x) is differentiable at x= a., (f), 00, If f (x) be any given function. Then,, Standard formula for limit, f' (a), [ LHD = RHD], at x a., %3D, Lt, x"- an, n - 1, = na, If f (x) is diff. at x = a., X-a, X → a, Then f (x) will also be continuous at x = a., But converse (Tl) is not true., Lt, sine, = 1, Polynomial function, f (x) = ax + bx + c, Trigonometric function, (iii) Lt, tane, = 1, f(x) = sinx, cos,. ., Modulus function., (iv) Lt, aX, = log a, f(x) = |x |, |x+ 1|,, Exponential function., Log, e, 1, %3D, f(x) = e%, 2*, a*,, Log 1, %3D, Logarithmic function., 1, f(x) = log, x,, sin (T, 2n, 3n, 4n,.., ) = 0, Rational function., cos (2π, 4π, 6π, 8π ., = 1, f(x) =, Q+ 0, cos (7, 3n, 57, 7n,.., ) = - 1, These function are always continuous in its domain., A 3n 5T, cos (, .) = 0, 2, If f(x) and g (x) are two continuous function., tan (r, 2n, 3n, 4.,.., .) = 0, Then, (i), f(x) + g (x), 3n 5n, tan (-, 2' 2, ) = 00, (ii) f (x) - g (x), 2, Differentiability of a function., (iii) f (x) x g (x), If f(x) be any given function then for differentiability, (iv) f (x)/ gx, at x = a, are also continuous., II