Tiêu đề 5. CONTINUITY & DIFFERENTIABILITY.pdf ✅

TOPIC-1 Continuity Concepts Covered:  Left hand Limit  Right Hand Limit Revision Notes FORMULAE FOR LIMITS: (a) lim cos x x → = 0 1 (b) lim sin x x → x = 0 1 (c) lim tan x x → x = 0 1 (d) lim sin x x → x − = 0 1 1 (e) lim tan x x → x − = 0 1 1 (f) lim log , x x e a x a a → − = > 0 1 0 (g) lim x x e → x − = 0 1 1 (h) lim log ( ) x e x → x + = 0 1 1 (i) lim x a n n x a n x a na → − − − = 1 z For a function f(x), lim x m → f(x) exists if lim x m → − f(x) = lim x m → + f(x). z A function f(x) is continuous at a point x = m if, lim ( ) lim ( ) ( ) xm xm fx fx f m → − → + = = , where lim ( ) x m f x → − is Left Hand Limit of f(x) at x = m and lim ( ) x m f x → + is Right Hand Limit of f(x) at x = m. Also f(m) is the value of function f(x) at x = m. [Board 2020] z A function f(x) is continuous at x = m (say) if, f(m) = lim x m → f(x), i.e., a function is continuous at a point in its domain if the limit value of the function at that point equals the value of the function at the same point. z For a continuous function f(x) atx = m, lim x m → f(x) can be directly obtained by evaluating f(m). z Indeterminate forms or meaningless forms: 0 0 0 1 00 0 , , , , , , . ∞ ∞ × ∞ ∞ − ∞ ∞ ∞ Example-1 Find the value of k for which the function f(x) = sin cos , , x x x x k x − − ≠ =       4 4 4 π π π is continuous at x = x 4 . Sol. lim sin cos x x x x f → − − =       π π π 4 4 4 lim sin cos x x x x k → −       − = π π 4 2 1 2 1 2 4 lim sin cos cos .sin x x x x k → −       − = π π π π 4 2 4 4 4 lim sin x x x k → −       −       = π π π 4 2 4 4 4 2 4 = k k = 1 2 2 UNIT – III : CALCULUS 5 CONTINUITY & DIFFERENTIABILITY CHAPTER Learning Objectives After going through this Chapter, the student would be able to:  Learn the continuity and differentiability of the function.  Relate continuity and differentiability.  Solve problems related to differentiability and continuity. LIST OF TOPICS Topic-1: Continuity Topic-2: Differentiability