Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsDirectional LimitsLimits and Continuity (§14.2)Math 241, Spring 2014Jayadev S. AthreyaSpring 2014Jayadev S. Athreya Math 241, Spring 2014Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsDirectional LimitsLimitsThe most important concept of calculus is the limit:Informally, we saylimx→af (x) = Lif no matter how close () you want f (x) to get to L, you can finda small neighborhood (of size δ) so that x being within δ of aguarantees that f (x) is within of L.All this relies on is a notion of distance.We have to be careful in higher dimensions, because you canapproach a point in many different ways.Jayadev S. Athreya Math 241, Spring 2014Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsDirectional LimitsOne-variable limitsThe sentence limx→af (x) = L means the following:For every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < |x − a| < δ implies |f (x) − L| < .Jayadev S. Athreya Math 241, Spring 2014Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsDirectional LimitsOne-variable limits: repriseGiven h ∈ R, h 6= 0, let E(h) = |f (a + h) −L|.The sentence limx→af (x) = L means the following:For every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < |h| < δ implies |E(h)| < .Jayadev S. Athreya Math 241, Spring 2014Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsDirectional LimitsOne-sided limitsIn one-variable, a limit exists if and only if the one-sided limitsexist and are equal.Limit from the left limx→a−f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < a − x < δ implies |f (x) − L| < .Limit from the right limx→a+f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < x − a < δ implies |f (x) − L| < .Jayadev S. Athreya Math 241, Spring 2014Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsDirectional LimitsOne-sided limitsIn one-variable, a limit exists if and only if the one-sided limitsexist and are equal.Limit from the left limx→a−f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < a − x < δ implies |f (x) − L| < .Limit from the right limx→a+f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < x − a < δ implies |f (x) − L| < .Jayadev S. Athreya Math 241, Spring 2014Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsDirectional LimitsOne-sided limitsIn one-variable, a limit exists if and only if the one-sided limitsexist and are equal.Limit from the left limx→a−f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < a − x < δ implies |f (x) − L| < .Limit from the right limx→a+f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < x − a < δ implies |f (x) − L| < .Jayadev S. Athreya Math 241, Spring 2014Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsDirectional LimitsOne-sided limits, repriseIn one-variable, a limit exists if and only if the one-sided limitsexist and are equal.Given h ∈ R, h 6= 0, let E(h) = |f (a + h) −L|.Limit from the left limx→a−f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < h < δ implies |E(a − h)| < .Limit from the right limx→a+f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < h < δ implies |E(a + h)| < .Jayadev S. Athreya Math 241, Spring 2014Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsDirectional LimitsOne-sided limits, repriseIn one-variable, a limit exists if and only if the one-sided limitsexist and are equal.Given h ∈ R, h 6= 0, let E(h) = |f (a + h) −L|.Limit from the left limx→a−f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < h < δ implies |E(a − h)| < .Limit from the right limx→a+f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < h < δ implies |E(a + h)| < .Jayadev S. Athreya Math 241, Spring 2014Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsDirectional LimitsOne-sided limits, repriseIn one-variable, a limit exists if and only if the one-sided limitsexist and are equal.Given h ∈ R, h 6= 0, let E(h) = |f (a + h) −L|.Limit from the left limx→a−f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < h < δ implies |E(a − h)| < .Limit from the right limx→a+f (x) = L ifFor every > 0 (acceptable error)There is δ > 0 (how close you need to be)Such that 0 < h < δ implies |E(a + h)| < .Jayadev S. Athreya Math 241, Spring 2014Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsDirectional LimitsExamples: linear functionLet’s try and see that limx→34x + 1 =13.Given h 6= 0 What is E(h) = |(4(3 + h) + 1) −13|? E(h) = 4|h|.Making E(h) small So to make E(h) < , what do we need tomake |h| smaller than?We get δ = /4.Jayadev S. Athreya Math 241, Spring 2014Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsDirectional LimitsExamples: linear functionLet’s try and see that limx→34x + 1 = 13.Given h 6= 0 What is E(h) = |(4(3 + h) + 1) −13|? E(h) = 4|h|.Making E(h) small So to make E(h) < , what do we need tomake |h| smaller than?We get δ = /4.Jayadev S. Athreya Math 241, Spring 2014Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsDirectional LimitsExamples: linear functionLet’s try and see that limx→34x + 1 = 13.Given h 6= 0 What is E(h) = |(4(3 + h) + 1) −13|?E(h) = 4|h|.Making E(h) small So to make E(h) < , what do we need tomake |h| smaller than?We get δ = /4.Jayadev S. Athreya Math 241, Spring 2014Limits and Continuity (§14.2)One-variable limitsHigher-dimensional limitsDirectional LimitsExamples: linear functionLet’s try and see that limx→34x + 1 = 13.Given h 6= 0 What is E(h) = |(4(3 + h) + 1) −13|? E(h) = 4|h|.Making E(h) small So to make E(h) < , what do we need
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