University:
University of Cambridge
Course:
0580 IGCSE | Mathematics
Academic year:

2023
Views:

488

Pages:

4
Author:

Shahbaz A.

Calculus and analytic geometry shahbaz ahmed August 2024 Introduction to the relevant concepts and skills The Left Hand Limit limitx→c− f (x) = L For values of x < c ,as x is getting close to c, the corresponding values 1 of f(x) are getting close to L. The Right Hand Limit limitx→c+ f (x) = M For values of x > c ,as x is getting close to c, the corresponding values of f(x) are getting close to M. Criterion for existence of Limit of a Function limitx→c f (x) = L ⇔ limitx→c− f (x) = limitx→c+ f (x) Criterion for Continuity of a Function at a Number A function is said to be continuous at a number ”c” ⇔ 2 (i) f(c) is defined (ii) limitx→c f (x) exists, (iii)) limitx→c f (x)=f(c) Discuss the continuity of f(x) at x=c; f (x) =     2x + 5 i f x ≤ 2     ,c = 2       4x + 1 i f x > 2 Solution limitx→2− f (x) = 2x + 5 limitx→2− f (x) = 2(2) + 5 = 9 limitx→2+ f (x) = 4x + 1 limitx→2+ f (x) = 4(2) + 1 = 9 Also f(x)= 2x+5 at x=2 =⇒ f (2) = 2(2) + 5 = 9 Since 3 limitx→2− f (x)= limitx→2+ f (x) = 9 =⇒ limitx→2 f (x) = 9 Also f(2)=9 Hence all the conditions for continuity at x=2 are satisfied function is continuous at c=2 as shown in figure below 4