lecture5_1 - 2.4 One-sided Limits and Limits at Infinity...

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Unformatted text preview: 2.4 One-sided Limits and Limits at Infinity Finite Limits as x Consider f (x) = 1 x as x What if x becomes increasingly large? What if x is negative and its magnitude becomes increasingly large? 1 becomes increasingly x We say is a limit of 1 x at infinity and negative infinity. Two basic limits a) b) limx k = 1 limx x = 1 Definition We say that f (x) has the limit L as x approaches infinity and write if, for every number > 0, there exists a corresponding number M such that for all x We say that f (x) has the limit L as x approaches minus infinity and write if, for every number > 0, there exists a corresponding number N such that for all x 2 What about our limit laws? The properties are just like the properties of finite limits. THEOREM 8 Limit Laws as x If L, M , and k are real numbers and x lim f (x) = L x lim g(x) = M , then 1. Sum Rule: 2. Difference Rule: 3. Product Rule: 4. Constant Multiple Rule: 5. Quotient Rule: M = 0 6. Power Rule: r and s are integers with no common factors, s = 0, then 3 Examples 5 a) limx( x - 2) = b) limt- 3t6 = 2 Limits at Infinity of Rational Functions Divide the numerator and denominator by the highest power of x in the denominator! a) limh 2h h+2h-3 = 3 -2 3 4 b) limt- (2-t) = t4 2 Horizontal Asymptotes Definition A line y = b is a horizontal asymptote of the graph of the function f (x) if either: Examples What is the asymptote of f (x) = 2x3 +2x-3 ? x3 -2 5 What is the asymptote of f (x) = ex? Example 11 Find 1 limx sin( x ) 6 Sandwich Theorem Revisited The Sandwich Theorem also holds for limits as x . Example 13 Using the Sandwich Theorem, find the horizontal asymptote of the curve f (x) = 2 + sin x . x 7 ...
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lecture5_1 - 2.4 One-sided Limits and Limits at Infinity...

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