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Unformatted text preview: Summary of Chapter 2 Suppose the interval I is a subset of Domain( f ). ( a, b ) is an open interval and [ a, b ] is a closed interval. ‘ f ( x ) defined near c ’ means ‘ f ( x ) defined on some ( a, b ) and c is in ( a, b ).’ Definitions  Limits § 2.2 lim x → a f ( x ) = L exists ⇐⇒ f ( x ) is arbitrarily close to L when x is sufficiently close to a , x 6 = a . § 2.2 lim x → a f ( x ) = L exists ⇐⇒ f ( x ) is arbitrarily close to L when x is sufficiently close to a , x < a . § 2.2 lim x → a + f ( x ) = L exists ⇐⇒ f ( x ) is arbitrarily close to L when x is sufficiently close to a , x > a . § 2.2 lim x → a f ( x ) = ∞ means f ( x ) is arbitrarily large when x is sufficiently close to a , x 6 = a . (=∞ ······ arbitrarily large negative ······ ) § 2.6 lim x →∞ f ( x ) = L means f ( x ) is arbitrarily close to L when x is sufficiently large. ( x → ∞ , ······ x is sufficiently large negative) § 2.6 lim x →∞ f ( x ) = ∞ means f ( x ) is arbitrarily large when x is sufficiently large. (similarly lim x →±∞ f ( x ) = ±∞ ) § 2.4 lim x → a f ( x ) = L exists if for every ² > 0 there is a δ > such that  f ( x ) L  < ² whenever 0 <  x a  < δ . § 2.4 lim x → a f ( x ) = L exists if for every ² > 0 there is a δ > such that  f ( x ) L  < ² whenever a δ < x < a . § 2.4 lim x → a + f ( x ) = L exists if for every ² > 0 there is a δ > such that  f ( x ) L  < ² whenever a < x < a + δ . § 2.4 lim x → a f ( x ) = ∞ if for every M > 0 there is a δ > 0 such that f ( x ) > M whenever 0 <  x a  < δ . (can replace a by a or a + ) § 2.4 lim x → a f ( x ) =∞ if for every M > 0 there is a δ > 0 such that f ( x ) < M whenever 0 <  x a  < δ . (can replace a by a or a + ) § 2.6 lim x →∞ f ( x ) = L if for every ² > 0 there is an N such that  f ( x ) L  < ² whenever x > N . (for x → ∞ take x < N ) Definitions  Asymptotes § 2.2 x = a is a vertical asymptote of y = f ( x ) if lim x → b f ( x ) = ±∞ where b = a , a + , a § 2.6 y = L is a horizontal asymptote of y = f ( x ) if lim x →∞ f ( x ) = L or lim x →∞ f ( x ) = L . 1 Definitions  Continuity § 2.5 f is continuous at a if lim x → a f ( x ) = f ( a ), otherwise f is discontinuous at a § 2.5 some discontinuities : removable, infinite, jump. § 2.5 f is continuous from the right at a if lim x → a + f ( x ) = f ( a ) § 2.5 f is continuous from the left at a if lim x → a f ( x ) = f ( a ) § 2.5 f is continuous on an interval I if f is continuous everywhere in I . Definitions  Derivatives § 2.8 Equation of tangent line to y = f ( x ) at a is y f ( a ) = f ( a )( x a ). (if f ( a ) exists) § 2.8 derivative of f ( x ) at a is f ( a ) = lim h → f ( a + h ) f ( a ) h = lim x → a f ( x ) f ( a ) x a if this exists at a ....
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This note was uploaded on 02/14/2011 for the course BIOL 1201 taught by Professor Wishtichusen during the Spring '07 term at LSU.
 Spring '07
 Wishtichusen
 Biology

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