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

2023
Views:

248

Pages:

4
Author:

Shahbaz A.

Q4 Calculus and analytic geometry 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) Q4 If f (x) =    x + 2, i f x ≤ −1   c + 2, i f x > −1 Find ”c” so that limx−→−1 f (x) exists. Solution limx−→−1− f (x) = limx−→−1− [x + 2] limx−→−1− f (x) = limx−→−1− [−1 + 2] = 1 Similarly limx−→−1+ f (x) = limx−→−1+ [c + 2] = c + 2 limx−→−1 f (x) exists ⇔ limx−→−1− f (x) = limx−→−1+ f (x) 1=c+2 Or 3 c+2=1 c=-1 4