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Chapter 2:Limits and Continuity. 2.1 Limits and Rates of Change. lim x→c f(x) = L. The real number L is a limit of the function f(x) as its x values approach the real number c if 0 < |x −c | < δ → | f(x) − L |< ε
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Chapter 2:Limits and Continuity 2.1 Limits and Rates of Change
limx→cf(x) = L • The real number L is a limit of the function f(x) as its x values approach the real number c if 0 < |x−c| < δ → | f(x) − L |< ε • All x values within a neighborhood δ units on either side of c will have corresponding y values within a neighborhood ε units on either side of L. (for positive δ and ε)
Theorem 3: 1- and 2-Sided Limits RIGHT-HAND limit: lim x→c+f(x) Value that f(x) approaches as x approaches c from the right side • LEFT-HAND limit: lim x→c-f(x) Value that f(x) approaches as x approaches c from the left side • A function has a limit as x approaches c if and only if the right-hand and left-hand limits both exist at x = c AND are equal.
Theorem 1 Properties of LimitsExamples Given:limx→cf(x) = 2 and limx→cg(x) = -3 • Sum : limx→c (f(x) + g(x)) = 2 + -3 = -1 • Difference : limx→c(f(x) - g(x)) = 2 - -3 = 5 • Product : limx→c(f(x) x g(x)) = 2 x -3 = -6 • Constant Multiple : limx→c(k x g(x))= -3k • Quotient Product : limx→c(f(x) ÷ g(x)) = -2/3 • Power : limx→c(f(x))r/s = 2r/s
Theorem 2:Polynomial and Rational Functions • For polynomial function f(x) and any real number c, lim x→cf(x) = f(c) • Ex. limx→-2(x² +3x +4) = (-2)² +3(-2) + 4 =2 • For rational function [f(x) ÷ g(x)], • limx→c [f(x)÷g(x)] = f(c)/g(c) for g(c)≠0 • Ex. limx→2 [(x²-1)÷(x+1)] = (4-1)/(2+1)=1 • Substitution works fine for most functions!
Limx→c f(x) exists • When f(x) is continuous at x = c (f(c)= the limit; the function takes on its limit value at x = c) • Or when f(x) has a REMOVEABLE (single point) discontinuity at x = c (The function may not exist at the point x = c or the function may take on a value other than the value of its limit at x = c.)
If a limit exists, then it can be algebraically evaluated • Some methods we will use in Section 2.1: • (1) by direct substitution Always try this FIRST! • (2) by algebraic manipulation • (3) by the Sandwich Theorem
Limx→c f(x) will fail to exist • if the right-hand and left-hand limits fail to both exist and to be equal at x = c • If f(x) has a vertical asymptote at x = c • If f(x) has a non-removeable discontinuity at x = c (jump, oscillating, or infinite)
