Intro+to+the+derivative+notes.pdf

The line x 2 is a vertical asymptote infinite limits

Info icon This preview shows pages 9–13. Sign up to view the full content.

View Full Document Right Arrow Icon
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 .
Image of page 9

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 .
Image of page 10
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!
Image of page 11

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 12
Image of page 13
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern