Ell defined also the functions we deal with must take

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: x = a we have two corresponding CHANGES δ y. f(x) which δ y do we choose ? δx x a a + δx (0, 0) We may make f(x) SINGLE VALUED by restricting the range as in either one of the diagrams below. f(x) f(x) δy δx δy δx (0, 0) a a + δx x (0, 0) 22 a a + δx x The function y = √x for x ≥ 0 is not SINGLE VALUED. It has two values: +√ x and − √x . We must specify which root we are using. y y = +√ x δx (0, 0) δy > 0 a+δx a δx x δy < 0 y = −√ x Since we are dealing with real valued functions over the real line we avoid cases real real where the function may take on complex values such as √x for x < 0. ell-defined Also, the functions we deal with must take on well-defined real values. We avoid undefined entities such as +∞, −∞, division by zero and ∞/∞. Functions of the form a n x n + a n-- 1 x n--1 + . . . + a 2 x 2 + a 1 x + a 0 a re called polynomials . These functions are single valued. p olynomials single 23 8. L I M I T o f a F u n c t i o n To understand the behaviour of a function f(x) very close to some instant a , that instant is to say near instant a and to the LEFT and RIGHT of instant a , we must instant instant analyse : L imit { f(a − δ x) } L imit { f(x) } or δx → 0 x → a-- L imit { f(x) } and x → a+ or L imit { f(a + δ x) } δx → 0 If it turns out that there is some definitive real number b such that : LIMIT from the LEFT = b = LIMIT from the RIGHT L imit { f(x) } =b= L imit { f(a − δ x) } =b= x → a-δx → 0 then we say : L imit { f(x) } x → a+ L imit { f(a + δ x) } δx → 0 L imit { f(x) } = b . x→a Properly speaking, finding or calculating the xLimit+ { f(x) } is a 2 step process. →a This becomes clear when we use the infinitesimal δ x. step 1. TENDS TO step : as x → a+ we may say x = a + δ x. We may substitute (a + δ x) for x in f(x). Then we do whatever expansion, regrouping of terms, cancellation and simplification possible. We may perform these operations because δ x ≠ 0 . It only TENDS TO zero. There is no division by...
View Full Document

This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.

Ask a homework question - tutors are online