Math131 Calculus I The Limit Laws Notes 2.3 I. The Limit Laws Assumptions: c is a constant and limf ( x ) and limg ( x ) exist x→ a x→ a Limit Law in symbols Limit Law in words The limit of a sum is equal to 1 lim[f ( x )+ g ( x )] = lim f ( x ) + lim g ( x ) x→ a x→ a x→ a the sum of the limits. The limit of a difference is equal to 2 lim[f ( x )− g ( x )] = lim f ( x ) − lim g ( x ) x→ a x→ a x→ a the difference of the limits. The limit of a constant times a function is equal 3 limcf ( x )= c lim f ( x ) x→ a x→ a to the constant times the limit of the function. The limit of a product is equal to 4 lim[f ( x ) g ( x )]= lim f ( x ) ⋅ lim g ( x )] x→ a x→ a x→ a the product of the limits. limf ( x ) f() x x→ a The limit of a quotient is equal to 5 lim = (iflim g ( x )≠ 0) x→ a g() x limg ( x ) x→ a the quotient of the limits. x→ a n n 6 lim[f ( x )]= [lim f ( x )] where n is a positive integer x→ a x→ a The limit of a constant function is equal 7 lim c= c x→ a to the constant. The limit of a linear function is equal 8 lim x= a x→ a to the number x is approaching. n n 9 lim x= a where n is a positive integer x→ a n n where n is a positive integer & if n is even, 10 lim x= a x→ a we assume that a > 0 where n is a positive integer & if n is even, n 11 limf ( x )= n lim f ( x ) x→ a x→ a we assume that limf ( x ) > 0 x→ a Direct Substitution Property: If f is a polynomial or rational function and a is in the domain of f, then limf ( x ) = x→ a “Simpler Function Property”: If f()() x= g x when x ≠ a then limf ( x )= lim g ( x ) , as long as the x→ a x→ a limit exists. Math131 Calculus I Notes 2.3 page 2 ex#1 Given limf ( x )= 2 , limg ( x )= − 1, limh ( x )= 3 use the Limit Laws find limf ( x ) h ( x )− x2 g ( x ) x→3 x→3 x→3 x→3 2x 2 + 1 ex#2 Evaluate lim , if it exists, by using the Limit Laws. x→2 x 2 + 6x − 4 ex#3 Evaluate: lim 2x2 + 3 x − 5 x→1 1− (1 − x )2 ex#4 Evaluate: lim x→0 x h +4 − 2 ex#5 Evaluate: lim h→0 h Math131 Calculus I Notes 2.3 page 3 Two Interesting Functions 1. Absolute Value Function x if x ≥ 0 Definition: x = −x if x < 0 Geometrically: The absolute value of a number indicates its distance from another number. x− c = a means the number x is exactly _____ units away from the number _____. x− c < a means: The number x is within _____ units of the number _____. How to solve equations and inequalities involving absolute value: Solve: |3x + 2| = 7 Solve: |x - 5| < 2 What does |x - 5| < 2 mean geometrically? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 2. The Greatest Integer Function Definition: [[x]] = the largest integer that is less than or equal to x. ex 6 [[ 5 ]] = ex 7 [[ 5.999 ]] = ex 8 [[ 3 ]] = ex 9 [[ -2.6 ]] = ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Theorem 1: limf ( x ) = L if and only if limf ( x )= L= lim f ( x ) x→ a x→ a− x→ a + ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ x ex#10 Prove that the lim does not exist. x→0 x Math131 Calculus I Notes 2.3 page 4 ex#11 What is lim[[x ]] ? x→ 3 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Theorem 2: If f( x )≤ g ( x ) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a then limf ( x )≤ lim g ( x ) . x→ a x→ a ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1 ex12 Find limx2 sin . To find this limit, let’s start by graphing it. Use your graphing calculator. x→0 x ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The Squeeze Theorem: If f()()() x≤ g x≤ h x when x is near a (except possibly at a) and limf ( x )= lim h ( x ) = L then limg ( x ) = L x→ a x→ a x→ a Math131 Calculus I Limits at Infinity & Horizontal Asymptotes Notes 2.6 Definitions of Limits at Large Numbers Definition in Words Precise Mathematical Definition Let f be a function defined on some interval (a, ∞). Let f be a function defined on some interval (a, ∞). Then limf ( x ) = L if for every ε > 0 there is a Then limf ( x ) = L means that the values of f(x) can x→∞ x→∞ Large be made arbitrarily close to L by taking x sufficiently corresponding number N such that if x > N then numbers POSITIVE large in a positive direction. f() x− L < ε Let f be a function defined on some interval Let f be a function defined on some interval (-∞,a). Then limf ( x ) = L if for every ε > 0 there (-∞,a). ∞). Then limf ( x ) = L means that the x→−∞ x→−∞ Large values of f(x) can be made arbitrarily close to L by is a corresponding number N such that if x < N then numbers NEGATIVE taking x sufficiently large in a negative direction. f() x− L < ε Definition What this can look like… The line y = L is a horizontal asymptote of the curve y = f(x) if either is true: 1. limf ( x ) = L x→∞ or 2. limf ( x )= L Horizontal Asymptote x→−∞ The line x = a is a vertical asymptote of the curve y = f(x) if at least one of the following is true: 1. limf ( x ) = ∞ x→ a 2. limf ( x ) = ∞ x→ a− 3. limf ( x ) = ∞ x→ a+ Vertical AsymptoteVertical 4. limf ( x ) = −∞ x→ a 5. limf ( x ) = −∞ x→ a− 6. limf ( x ) = −∞ x→ a+ Theorem 1 • If r > 0 is a rational number then lim = 0 x→∞ x r 1 • If r > 0 is a rational number such that x r is defined for all x then lim = 0 x→−∞ x r Math131 Calculus I Notes 2.6 page 2 3 ex#1 Find the limit: lim x→∞ x 5 x3 + 2 x ex#2 Find the limit: lim x→ ∞ 5 x 3−x 2 + 4 ex#3 Find the limit: lim 9x2 + x − 3 x x→ ∞ ex#4 Find the limit: lim cos x x→∞ Math131 Calculus I Notes 2.6 page 3 2x 2 + 1 ex#5 Find the vertical and horizontal asymptotes of the graph of the function: f() x = 3x − 5 , y !!$rP$L 1. o\+{-i - g r4-,+-CR.\ r: orot a\r.eA \o\k-i \$e. dov oq - -\ob ^g FroA\..cf - . 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