alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

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

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: + { 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...
View Full Document

Ask a homework question - tutors are online