alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# We can simplify first because irst take simplify ir a

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Unformatted text preview: he RIGHT a L I M I T { f(x) } = L I M I T { x -- a } = L I M I T { 1 } = 1 x -x → a+ x → a+ x → a+ 1. TENDS TO step : near a and to the right of a : x = a + δx right +δ x f(x) = { (a +δ x) -- a } = { }=1 +δ x (a +δ x) -- a irst take Note how we simplify ffirst and then take the limit. We can simplify first because simplify ir δx only TENDS TO zero. δx ≠ 0. So there is no division by zero in this step. division by zer ero step So there 2. LIMIT step : L I M I T {1} = 1 δx → 0 Since, L I M I T − { f(x) } = 1 = L I M I T + { f(x) } , we say L I M I T { f(x) } = 1. x→a x→a x→a VALUE { f(x) } = VALUE { x -- a } = 0 0 , which is something undefined. / x=a x = a x -- a 28 2 2 Example 5: What is the LIMIT of f(x) = x -- a as x TENDS TO a ? x -- a Limit from the LEFT L imit {f(x)} = L imit { x2 -- a2 } = L imit {(x--a)(x+a)} = L imit {x+a} = 2a x -- a x → a-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 2 -- a2 -- 2a.δ x + δ x 2 (a -- δ x) f(x) = { }={ } = 2a -- δ x (a -- δ x) -- a -- δ x irst take Note how we simplify ffirst and then take the limit. We can simplify first because simplify ir δx only TENDS TO zero. δx ≠ 0. There is no division by zero in this step. here division by zer ero step 2. LIMIT step : L imit → { 2a -- δ x } = 2a Limit from the RIGHT 2 2 L imit {f(x)} = L imit { x -- a } = L imit { (x--a)(x+a)} = L imit {x+a} = 2a x -- a x -- a x → a+ x → a+ x → a+ x → a+ δx 0 1. TENDS TO step : near a and to the right of a : right x = a + δx (a + δ x)2 -- a2 2a.δ x+ δ x 2 }= { } = 2a + δ x (a + δ x) -- a δx 2. LIMIT step : L imit { 2a + δ x } = 2a → f(x) = { δx 0 Since, L imit-- { f(x) } = 2a = L imit+ { f(x) } , we say x → a-- x → a+ L imit { f(x) } = 2a. x→a Value { f(x) } = Value { x2 -- a2 } = 0 , which is somet...
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