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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 > 0 there is a > such that  f ( x ) L  < whenever 0 <  x a  < . 2.4 lim x a f ( x ) = L exists if for every > 0 there is a > such that  f ( x ) L  < whenever a < x < a . 2.4 lim x a + f ( x ) = L exists if for every > 0 there is a > such that  f ( x ) L  < whenever a < x < a + . 2.4 lim x a f ( x ) = if for every M > 0 there is a > 0 such that f ( x ) > M whenever 0 <  x a  < . (can replace a by a or a + ) 2.4 lim x a f ( x ) = if for every M > 0 there is a > 0 such that f ( x ) < M whenever 0 <  x a  < . (can replace a by a or a + ) 2.6 lim x f ( x ) = L if for every > 0 there is an N such that  f ( x ) L  < 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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 Spring '07
 Wishtichusen
 Biology

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