Intro+to+the+derivative+notes.pdf

The line x 2 is a vertical asymptote infinite limits

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The line x = 2 is a vertical asymptote. Infinite Limits Definitions 3.5. Let f be defined on both sides of a real number a , except possible at a itself. Then lim x a f ( x ) = means that the values of f ( x ) can be made arbitrarily large (as large as we please) by taking x sufficiently closed to a , but not equal to a . lim x a f ( x ) = -∞ means that the values of f ( x ) can be made arbi- trarily large negative by taking x sufficiently closed to a , but not equal to a .
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8 J. S´ anchez-Ortega Similar definitions can be given for the one-sided infinite limits lim x a - f ( x ) = lim x a + f ( x ) = lim x a - f ( x ) = -∞ lim x a + f ( x ) = -∞ Definition 3.6. The line x = a is called a vertical asymptote of the curve y = f ( x ) if at least one of the following statements is true: lim x a f ( x ) = lim x a - f ( x ) = lim x a + f ( x ) = lim x a f ( x ) = -∞ lim x a - f ( x ) = -∞ lim x a + f ( x ) = -∞ Example 3.7. As we could deduce from the graph of the tangent function, its vertical asymptotes are x = π 2 + , where k Z .
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3. Introduction to the derivative: Limits and Continuity 9 Limits at Infinity Definition 3.8. If the function f us defined on an interval ( a, + ) and if we can ensure that f ( x ) is as close as we want to the number L by taking x large enough, then we say that f ( x ) approaches the limit L as x approaches infinity , and we write lim x →∞ f ( x ) = L . In such a case, we will say that the line y = L is a horizontal asymptote of the curve y = f ( x ). If f is defined on an interval ( -∞ , b ) and if we can ensure that f ( x ) is as close as we want to the number M by taking x negative and large enough, then we say that f ( x ) approaches the limit M as x approaches negative infinity , and we write lim x →-∞ f ( x ) = M . In such a case, we will say that the line y = M is a horizontal asymptote of the curve y = f ( x ). Example 3.9. Let f ( x ) = 1 /x 2 . Looking at the graph, we could see that lim x + 1 x 2 = 0 , lim x →-∞ 1 x 2 = 0 In this case the x -axis is a horizontal asymptote of the curve y = 1 /x 2 . See also Example 3 (page 693) in Textbook!
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10 J. S´ anchez-Ortega 3.2 Limit Laws Let f , g be two functions and c a constant. Assume that the limits lim x a f ( x ) and lim x a g ( x ) exist. Then L1. lim x a [ f ( x ) + g ( x )] = lim x a f ( x ) + lim x a g ( x ) L2. lim x a [ f ( x ) - g ( x )] = lim x a f ( x ) - lim x a g ( x ) L3. lim x a [ cf ( x )] = c lim x a f ( x ) L4. lim x a [ f ( x ) g ( x )] = lim x a f ( x ) · lim x a g ( x ) L5. lim x a f ( x ) g ( x ) = lim x a f ( x ) lim x a g ( x ) if lim x a g ( x ) 6 = 0 L6. lim x a [ f ( x )] n = lim x a f ( x ) n L7. lim x a c = c L8. lim x a x = a L9. lim x a x n = a n where n is a positive integer L10. lim x a n x = n x where n is a positive integer (If n is even, we assume that a > 0) L11. lim x a n p f ( x ) = n p lim x a f ( x ) where n is a positive integer (If n is even, we assume that lim x a f ( x ) > 0) L12. Direct Substitution Property. If f is a polynomial or a rational function and a is in the domain of f , then lim x a f ( x ) = f ( a ) This property tells us that polynomials and rational functions are con- tinuous functions on their respective domains.
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