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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. elldefined
Also, the functions we deal with must take on welldefined real values. We avoid
undefined entities such as +∞, −∞, division by zero and ∞/∞.
Functions of the form a n x n + a n 1 x n1 + . . . + 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 → aδx → 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...
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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.
 Fall '09
 TAMERDOğAN

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