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MTH 251 – Differential Calculus Chapter 2 Review Limits and Continuity Copyright © 2010 by Ron Wallace, all rights reserved. Calculating Limits - 1 lim f ( x ) x c • Polynomial: f(c) • Rational: f(c) if there is no division by 0 • Radical: f(c) if there are no even roots of negatives. • Exponential: f(c) • Logarithmetic: f(c) if there a no logs of nonpositives • Trigonometric: f(c) except where the trig functions are undefined. • Sums, Differences, Products, Quotients (except division by zero), and Compositions of these. Calculating Limits - 2 f ( x) lim x c g ( x ) m f ( x) lim x c g ( x ) m n 0 f ( x) lim x c g ( x ) f ( x) lim x c g ( x ) Note: Assume m 0 & n 0 0 n 0 0 Calculating Limits - 2 x 5 lim x 2 x 2 x 9 lim 2 x 3 x 3 5x 1 x 5 lim x 5 x 5 2 lim x 1 x 1 2 2 Calculating Limits - 2 f ( x) lim x c g ( x ) 0 0 • Algebraic equivalences. Remove (x – c) if it is a common factor. Rationalize the numerator or denominator Calculating Limits - 2 f ( x) lim x c g ( x ) 0 0 • Algebraic equivalences. Remove (x – c) if it is a common factor. Rationalize the numerator or denominator x 3x 10 lim 2 x 2 x 4 2 Calculating Limits - 2 f ( x) lim x c g ( x ) 0 0 • Algebraic equivalences. Remove (x – c) if it is a common factor. Rationalize the numerator or denominator x 5 3 lim x4 x4 Calculating Limits - 3 sin mx lim x 0 mx 1 lim x x ax lim n x bx m mn mn mn Calculating Limits - 4 f ( x) lim x c g ( x ) f ( x) lim x c g ( x ) n 0 f ( x) lim x c g ( x ) m f ( x) lim x c g ( x ) 0 Note: Assume m 0 & n 0 Calculating Limits - 5 lim f ( x ) x c • Left-Hand Limit • Only need to consider x < c lim f ( x ) x c • Right-Hand Limit • Only need to consider x > c Calculating Limits – 3, 4, & 5 3x 11 lim 2 x 5 x 7 x 2 lim x 5 7x x 5 sin 2 x lim x 0 x cos x cos 3x lim tan x x 2 When a limit DNE lim f ( x ) DNE x c • f(x) is not defined around c • Jump • Usually a piecewise function • Oscillation • Usually involves sine or cosine • Increase/Decrease without bound • Vertical asymptotes • May be different on left and right Proving Limits • Definition lim f ( x ) L x c f(x) if, for every > 0, there exists a > 0 such that … 0 xc f ( x) L L+ L if, 0 0 0 x c f ( x) L L- c- c c+ You MUST be able to state this & draw the diagram w/ labels. Proving Limits • Process … Begin with f ( x) L • i.e. f ( x ) L lim f ( x ) L x c may or may not be given Manipulate to get … a x c b Determine min a , b note: a 0 & b 0 Proving Limits • Prove that … lim x 5 3 3 x 2 using 0.01 Round calculations to 5 decimal places. Continuity • f(x) is continuous at x = c if and only if … When c is an interior point of the domain and lim f ( x ) f ( c ) x c When c is a left endpoint of the domain and lim f ( x ) f (c) x c When c is a right endpoint of the domain and lim f ( x ) f (c ) x c NOTE: You MUST be able to state this definition. Continuous Functions • A function that is continuous for all values of its domain. • All of the “elementary functions” are continuous functions. polynomials, rationals, radicals, exponentials, logarithms, absolute values, trigonometric, and combinations of these – note: Consider the domains Common Points of Discontinuity • f(c) is not defined division by zero square roots of negatives asymptotes of trigonometric functions logarithms of non-positives • The limit as x c DNE oscillations • f(c) does not equal the limit as x c piece-wise functions where the endpoints of the pieces don’t connect Discontinuity Examples • Where are these functions NOT continuous? f ( x) x 6 x 5 3 x5 f ( x) 2 x 25 x 2 +1, x 0 f ( x) 2 x 1, x 0 4 x 2 1