alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

We can simplify first because irst take simplify ir a

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online