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# Lesson 3: The Concept of Limit

## by Matthew Leingang, Clinical Associate Professor of Mathematics at New York University on Jan 26, 2010

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The limit is how we describe functions near points.

The limit is how we describe functions near points.

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## Lesson 3: The Concept of LimitPresentation Transcript

• Section 1.3 The Concept of Limit V63.0121.006/016, Calculus I January 26, 2009 Announcements Blackboard sites are up Ofﬁce Hours: MW 1:30–2:30, R 9–10 (CIWW 726) WebAssignments not due until Feb 2 (but there are several) . . . . . .
• Limit . . . . . .
• Zeno’s Paradox That which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics VI:9, 239b10) . . . . . .
• Outline Heuristics Errors and tolerances Examples Pathologies Precise Deﬁnition of a Limit . . . . . .
• Heuristic Deﬁnition of a Limit Deﬁnition We write lim f(x) = L x→a and say “the limit of f(x), as x approaches a, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufﬁciently close to a (on either side of a) but not equal to a. . . . . . .
• Outline Heuristics Errors and tolerances Examples Pathologies Precise Deﬁnition of a Limit . . . . . .
• The error-tolerance game A game between two players to decide if a limit lim f(x) exists. x→a Step 1 Player 1: Choose L to be the limit. Step 2 Player 2: Propose an “error” level around L. Step 3 Player 1: Choose a “tolerance” level around a so that x-points within that tolerance level of a are taken to y-values within the error level of L, with the possible exception of a itself. Step 4 Go back to Step 2 until Player 1 cannot move. If Player 1 can always ﬁnd a tolerance level, lim f(x) = L. x →a . . . . . .
• The error-tolerance game L . . a . . . . . . .
• The error-tolerance game L . . a . . . . . . .
• The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . .
• The error-tolerance game T . his tolerance is too big L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . .
• The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . .
• The error-tolerance game S . till too big L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . .
• The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . .
• The error-tolerance game T . his looks good L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . .
• The error-tolerance game S . o does this L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . .
• The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. If Player 2 shrinks the error, Player 1 can still win. . . . . . .
• The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. If Player 2 shrinks the error, Player 1 can still win. . . . . . .
• Outline Heuristics Errors and tolerances Examples Pathologies Precise Deﬁnition of a Limit . . . . . .
• Example Find lim x2 if it exists. x →0 . . . . . .
• Example Find lim x2 if it exists. x →0 Solution I claim the limit is zero. . . . . . .
• Example Find lim x2 if it exists. x →0 Solution I claim the limit is zero. If the error level is 0.01, I need to guarantee that −0.01 < x2 < 0.01 for all x sufﬁciently close to zero. . . . . . .
• Example Find lim x2 if it exists. x →0 Solution I claim the limit is zero. If the error level is 0.01, I need to guarantee that −0.01 < x2 < 0.01 for all x sufﬁciently close to zero. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so I win that round. . . . . . .
• Example Find lim x2 if it exists. x →0 Solution I claim the limit is zero. If the error level is 0.01, I need to guarantee that −0.01 < x2 < 0.01 for all x sufﬁciently close to zero. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so I win that round. What should the tolerance be if the error is 0.0001? . . . . . .
• Example Find lim x2 if it exists. x →0 Solution I claim the limit is zero. If the error level is 0.01, I need to guarantee that −0.01 < x2 < 0.01 for all x sufﬁciently close to zero. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so I win that round. What should the tolerance be if the error is 0.0001? By setting tolerance equal to the square root of the error, we can guarantee to be within any error. . . . . . .
• Example |x| Find lim if it exists. x →0 x . . . . . .
• Example |x| Find lim if it exists. x →0 x Solution The function can also be written as { |x| 1 if x > 0; = x −1 if x < 0 What would be the limit? . . . . . .
• The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
• The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
• The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
• The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
• The error-tolerance game y . . . 1 . x . . Part of graph in- . 1. − side blue is not inside green . . . . . .
• The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
• The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
• The error-tolerance game y . . Part of graph in- side blue is not . . 1 inside green . x . . 1. − . . . . . .
• The error-tolerance game y . . Part of graph in- side blue is not . . 1 inside green . x . . 1. − These are the only good choices; the limit does not exist. . . . . . .
• One-sided limits Deﬁnition We write lim f(x) = L x →a + and say “the limit of f(x), as x approaches a from the right, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufﬁciently close to a and greater than a. . . . . . .
• One-sided limits Deﬁnition We write lim f(x) = L x →a − and say “the limit of f(x), as x approaches a from the left, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufﬁciently close to a and less than a. . . . . . .
• The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
• The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
• The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
• The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
• The error-tolerance game y . . . 1 . x . . Part of graph in- . 1. − side blue is in- side green . . . . . .
• The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
• The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
• The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
• The error-tolerance game y . . Part of graph in- . . 1 side blue is in- side green . x . . 1. − . . . . . .
• The error-tolerance game y . . Part of graph in- . . 1 side blue is in- side green . x . . 1. − So lim f(x) = 1 and lim f(x) = −1 x→0+ x →0 − . . . . . .
• Example |x| Find lim if it exists. x →0 x Solution The function can also be written as { |x| 1 if x > 0; = x −1 if x < 0 What would be the limit? The error-tolerance game fails, but lim f(x) = 1 lim f(x) = −1 x →0 + x →0 − . . . . . .
• Example 1 Find lim if it exists. x →0 + x . . . . . .
• The error-tolerance game y . .? . L . x . 0 . . . . . . .
• The error-tolerance game y . .? . L . x . 0 . . . . . . .
• The error-tolerance game y . .? . L . x . 0 . . . . . . .
• The error-tolerance game y . . The graph escapes the green, so no good .? . L . x . 0 . . . . . . .
• The error-tolerance game y . .? . L . x . 0 . . . . . . .
• The error-tolerance game y . E . ven worse! .? . L . x . 0 . . . . . . .
• The error-tolerance game y . . The limit does not exist because the function is unbounded near 0 .? . L . x . 0 . . . . . . .
• Example 1 Find lim if it exists. x →0 + x Solution The limit does not exist because the function is unbounded near 0. Next week we will understand the statement that 1 lim = +∞ x →0 + x . . . . . .
• Weird, wild stuff Example (π ) Find lim sin if it exists. x →0 x . . . . . .
• Function values x π/x sin(π/x) . /2 π 1 π 0 . 1 /2 2π 0 1/k kπ 0 2 π/2 1 2 /5 5π/2 1 . . . .. π 0 2 /9 9π/2 1 2/13 13π/2 1 2 /3 3π/2 −1 2 /7 7π/2 −1 . 2/11 11π/2 −1 3 . π/2 . . . . . .
• Weird, wild stuff Example (π ) Find lim sin if it exists. x →0 x . . . . . .
• Weird, wild stuff Example (π ) Find lim sin if it exists. x →0 x f(x) = 0 when x = f(x) = 1 when x = f(x) = −1 when x = . . . . . .
• Weird, wild stuff Example (π ) Find lim sin if it exists. x →0 x 1 f(x) = 0 when x = for any integer k k f(x) = 1 when x = f(x) = −1 when x = . . . . . .
• Weird, wild stuff Example (π ) Find lim sin if it exists. x →0 x 1 f(x) = 0 when x = for any integer k k 2 f(x) = 1 when x = for any integer k 4k + 1 f(x) = −1 when x = . . . . . .
• Weird, wild stuff Example (π ) Find lim sin if it exists. x →0 x 1 f(x) = 0 when x = for any integer k k 2 f(x) = 1 when x = for any integer k 4k + 1 2 f(x) = −1 when x = for any integer k 4k − 1 . . . . . .
• Weird, wild stuff continued Here is a graph of the function: y . . . 1 . x . . 1. − There are inﬁnitely many points arbitrarily close to zero where f(x) is 0, or 1, or −1. So the limit cannot exist. . . . . . .
• Outline Heuristics Errors and tolerances Examples Pathologies Precise Deﬁnition of a Limit . . . . . .
• What could go wrong? Summary of Limit Pathologies How could a function fail to have a limit? Some possibilities: left- and right- hand limits exist but are not equal The function is unbounded near a Oscillation with increasingly high frequency near a . . . . . .
• Meet the Mathematician: Augustin Louis Cauchy French, 1789–1857 Royalist and Catholic made contributions in geometry, calculus, complex analysis, number theory created the deﬁnition of limit we use today but didn’t understand it . . . . . .
• Outline Heuristics Errors and tolerances Examples Pathologies Precise Deﬁnition of a Limit . . . . . .
• Precise Deﬁnition of a Limit No, this is not going to be on the test Let f be a function deﬁned on an some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L, and we write lim f(x) = L, x →a if for every ε > 0 there is a corresponding δ > 0 such that if 0 < |x − a| < δ , then |f(x) − L| < ε. . . . . . .
• The error-tolerance game = ε, δ L . . a . . . . . . .
• The error-tolerance game = ε, δ L . +ε L . . −ε L . a . . . . . . .
• The error-tolerance game = ε, δ L . +ε L . . −ε L . aa . − δ. . + δ a . . . . . .
• The error-tolerance game = ε, δ T . his δ is too big L . +ε L . . −ε L . aa . − δ. . + δ a . . . . . .
• The error-tolerance game = ε, δ L . +ε L . . −ε L . a . a+ . −. δ δ a . . . . . .
• The error-tolerance game = ε, δ T . his δ looks good L . +ε L . . −ε L . a . a+ . −. δ δ a . . . . . .
• The error-tolerance game = ε, δ S . o does this δ L . +ε L . . −ε L . a aa .+ . .− δ δ . . . . . .