Limits and Continuity Handout PDF

ITSH2302 Definition 1.1. lim 𝑓(𝑥) = 𝐿 if and only if ∀ 𝜀 > 0, ∃ a real 𝛿 > 0 such that whenever 0 < |𝑥 − 𝑐| < 𝛿, then 𝑥→𝑐 |𝑓(𝑥) − 𝐿| < 𝜀. Definition 1.2 We say that 𝐿 is the left-hand limit of 𝑓(𝑥) at 𝑐, written lim 𝑓(𝑥) = 𝐿...

ITSH2302 Definition 1.1. lim 𝑓(𝑥) = 𝐿 if and only if ∀ 𝜀 > 0, ∃ a real 𝛿 > 0 such that whenever 0 < |𝑥 − 𝑐| < 𝛿, then 𝑥→𝑐 |𝑓(𝑥) − 𝐿| < 𝜀. Definition 1.2 We say that 𝐿 is the left-hand limit of 𝑓(𝑥) at 𝑐, written lim 𝑓(𝑥) = 𝐿 𝑥→𝑐 − if ∀ 𝜀 > 0, ∃ a real 𝛿 > 0 such that whenever 𝑐 − 𝛿 < 𝑥 < 𝑐, |𝑓(𝑥) − 𝐿| < 𝜀. We say that 𝐿 is the right-hand limit of 𝑓(𝑥) at 𝑐, written lim 𝑓(𝑥) = 𝐿 𝑥→𝑐 + if ∀ 𝜀 > 0, ∃ a real 𝛿 > 0 such that whenever 𝑐 < 𝑥 < 𝑐 + 𝛿, |𝑓(𝑥) − 𝐿| < 𝜀. Definition 1.3 If a function 𝑓 increases or decreases without bound as 𝑥 approaches a real number 𝑐 from either the right or the left, then 𝑓 has a vertical asymptote at 𝑥 = 𝑐. A function has a vertical asymptote at 𝑥 = 𝑐 if one of the following conditions are satisfied: a. lim 𝑓(𝑥) = ∞ and lim− 𝑓(𝑥) = ∞ 𝑥→𝑐 + 𝑥→𝑐 b. lim+ 𝑓(𝑥) = ∞ and lim− 𝑓(𝑥) = −∞ 𝑥→𝑐 𝑥→𝑐 c. lim+ 𝑓(𝑥) = −∞ and lim− 𝑓(𝑥) = ∞ 𝑥→𝑐 𝑥→𝑐 d. lim+ 𝑓(𝑥) = −∞ and lim− 𝑓(𝑥) = −∞ 𝑥→𝑐 𝑥→𝑐 Definition 1.4 If the values of a function 𝑓(𝑥) approach a real number 𝐿 as 𝑥 increases or decreases without bound, then 𝑓 has a horizontal asymptote at 𝑦 = 𝐿. A non-constant function has a horizontal asymptote at 𝑦 = 𝐿 if one of the following conditions are satisfied: a. lim 𝑓(𝑥) = 𝐿 𝑥→∞ b. lim 𝑓(𝑥) = 𝐿 𝑥→∞− References: Azad, K. Better Explained. (n.d.). AA Gentle Introduction to Learning Calculus. Retrieved from: http://betterexplained.com/articles/a-gentle-introduction-to-learning-calculus/ last May 18, 2016. Coburn, J. (2016). Pre-Calculus. McGraw Hill Education. Minton, R. & Smith, R. (2016). Basic Calculus. McGraw Hill Education. 01 Handout 1 *Property of STI Page 1 of 1