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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# Limit step l imit a2 2a x x 2 a2 x 0 since

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Unformatted text preview: + { f(x) } , we say L imit { f(x) } = 2a . → x x→a a VALUE { f(x) } = f(a) = 2a . x=a Here we have : L I M I T { f(x) } = 2a = VALUE { f(x) } . x=a x→a Now see the graph on page 180. 26 Example 3: What is the LIMIT of f(x) = x2 as x TENDS TO a ? Limit from the LEFT L imit { f(x) } = L imit { x2 } = L imit-- { a2 } = a2 -→ x → a-- x x→a a Let us find L imit -- {f(x)} in 2 steps using δx. → x a left 1. TENDS TO step : near a and to the left of a : x = a − δx f(x) = (a − δ x)2 = a2 − 2aδ x + δ x 2 2. LIMIT step : L imit {a2 − 2aδ x + δ x 2 } = a2 δx → 0 Limit from the RIGHT L imit { f(x) } = L imit x → a+ x → a+ { x2 } = L imit x → a+ { a 2 } = a2 Let us find L imit + {f(x)} in 2 steps using δx. x→a 1. TENDS TO step : near a and to the right of a : right x = a + δx f(x) = (a + δ x)2 = a2 + 2aδ x + δ x 2 2. LIMIT step : L imit {a2 + 2aδ x + δ x 2 } = a2 δx → 0 Since, L imit-- { f(x) } = a2 = L imit+ { f(x) } , we say L imit { f(x) } = a2 . x→a Here we have : x→a x→a VALUE { f(x) } = f(a) = a2 . x=a L I M I T { f(x) } = a2 = VALUE { f(x) } . x=a x→a We should begin to get the general feeling that for polynomials of the form : polynomials pol f(x) = a n x n + a n-- 1 x n--1 + . . . + a 2 x 2 + a 1 x + a 0 we have: L I M I T { f(x) } = VALUE { f(x) } . x→a x=a 27 a Example 4: What is the LIMIT of f(x) = x -- a as x TENDS TO a ? x -Limit from the LEFT L I M I T { f(x) } = L I M I T { x -- a } = L I M I T { 1 } = 1 x → a− x -- a x → a− x → a− Note how we simplify ffirst and then take the limit. We can simplify first because irst take simplify ir − a) ≠ 0. Let us do this in 2 steps using δx. when x → a we have (x 1. TENDS TO step : near a and to the left of a : x = a − δx left -- δ x (a -- δ x) -- a f(x) = { }= { }=1 -- δ x (a -- δ x) -- a Note how we simplify ffirst and then take the limit. We can simplify first because irst take simplify ir division by zer ero step δx only TENDS TO zero. δx ≠ 0. So there is no division by zero in this step. So there 2. LIMIT step : L I M I T {1} = 1 δx → 0 Limit from t...
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