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

2023
Views:

190

Pages:

5
Author:

Shahbaz A.

Calculus and analytic geometry Q3 shahbaz ahmed August 2024 Introduction to the relevant concepts and skills The Left Hand Limit limx→c− f (x) = L 1 For values of x < c ,as x is getting close to c, the corresponding values of f(x) are getting close to L. The Right Hand Limit limx→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 limx→c f (x) = L ⇔ limx→c− f (x) = limx→c+ f (x) Criterion for Continuity of a Function at a Number 2 A function is said to be continuous at a number ”c” ⇔ (i) f(c) is defined (ii) limx→c f (x) exists, (iii)) limx→c f (x)=f(c) Q If     3x i f x ≤ −2     f (x) = x2 − 1 i f − 2 < x < 2       3 i f x ≥ 2 Discuss continuity at x=2 and x=-2 Solution limx→2− [ f (x)] = limx→2− [x2 − 1] limx→2− [ f (x)] = limx→2− [(2)2 − 1] = 3 Similarly limx→2+ [ f (x)] = limx→2+ [3] = 3 Since limx→2− [ f (x)]=limx→2+ [ f (x)] = 3 3 Therefore limx−→2 f (x) = 2 Also f (2) = 3 Since all conditions for continuity are satisfied, f(x) is continuous at x=2 Similarly limx→−2− [ f (x)] = limx→−2− [3x] limx→2− [ f (x)] = limx→2− [3(−2)] = −6 Similarly limx→−2+ [ f (x)] = limx→−2+ [x2 − 1] limx→−2+ [ f (x)] = limx→−2+ [(−2)2 − 1] = 3 Since limx→−2− [ f (x)] ̸= limx→−2+ [ f (x)] limx−→−2 f (x) does not exist f(x) is discontinuous at x=-2 Continuity at x=2 and discontinuity at x=-2 are shown in figure below 4 5