Ex 13.1, 30 - If f(x) = { |x| + 1, 0, |x| - 1 - Chapter 13 - Limits - Limit exists

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  1. Chapter 13 Class 11 Limits and Derivatives
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Ex 13.1, 30 If f(x) = x +1, x<0 0 x=0 x 1, x>0 . For what value (s) of a does f(x) exists? We need to find value of a for which lim x a f(x) exists lim x a f(x) exists only if lim x a + f (x) = lim x a f(x) We check limit different values of a When a = 0 When a < 0 When a > 0 Case 1: When a = 0 Limit exists at a = 0 if lim x 0 + f(x) = lim x 0 f(x) f(x) = x +1, x<0 0 x=0 x 1, x>0 . Since 1 1 lim x 0 f(x) lim x 0 + f(x) So, left hand limit and right hand limit are not equal at x = 0 Hence lim x 0 f(x) does not exists At x = 0 Limit does not exists Case 2: When x = a, a > 0 Limit exists at a if lim x + f(x) = lim x f(x) f(x) = x +1, x<0 0 x=0 x 1, x>0 . Since lim x f(x) = lim x + f(x) Hence at all point x = a, a > 0 f(x) exists Case 3: When x = a, a < 0 Limit exists at a if lim x + f(x) = lim x f(x) f(x) = x +1, x<0 0 x=0 x 1, x>0 . Since lim x f(x) = lim x + f(x) Hence at all point x = a, a < 0 f(x) exists f(x) exists exists for all a 0

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