1110-fa2011-PRELIM-1-SOLUTIONS

For the function to be continuous at x the limit lim

This preview shows page 5 - 6 out of 6 pages.

For the function to be continuous at x = 0 , the limit ( lim x 0 f ( x ) must exist and equal f ( 0 ) . Using the previous part, the limit exists if the left-hand and right-hand limits are equal, so a 2 = 4 , or a = ± 2 . We can check, f ( 0 ) = ( 0 ± 2 ) 2 = 4 , and so for either value, f ( 0 ) = 4 . Thus, the function f ( x ) is continuous for a = 2 or a = - 2 . Question 6. (20 points) Consider the function f ( x ) = x 2 x 2 - 4 . (a) Is f ( x ) an even function, an odd function, or neither? Please justify your answer. The function is even because f (- x ) = (- x ) 2 (- x ) 2 - 4 = x 2 x 2 - 4 = f ( x ) . (b) Compute lim x →∞ f ( x ) . What does this limit tell you about the asymptotes of the graph of f ( x ) ? We divide by x 2 in the numerator and denominator: lim x →∞ x 2 x 2 - 4 = lim x →∞ 1 1 - 4/x 2 = 1. This means that the graph of f ( x ) has a horizontal asymptote to the line y = 1 on the right hand side of the graph. The function is even, so it has symmetry across the y -axis, so it also has a horizontal asymptote on the left-hand side. (c) Find lim x 2 + f ( x ) . What does this limit tell you about the asymptotes of the graph of f ( x ) ? We separate out the terms: lim x 2 + x 2 x 2 - 4 = lim x 2 + x 2 x + 2 1 x - 2 . The limit lim x 2 + x 2 x + 2 = 1 . On the other hand, lim x 2 + 1 x - 2 = because the denominator will be a very small positive number, so its reciprocal will be a very large positive number. Therefore, the product must have lim x 2 + x 2 x 2 - 4 = lim x 2 + x 2 x + 2 1 x - 2 = . (Alternately, we can evaluate f ( x ) for x close to, but larger than, 2 , and see that we get a very large positive number.) This means that the graph of f ( x ) has an vertical asymptote to the line x = 2 . (d) Find lim x 2 - f ( x ) . What does this limit tell you about the asymptotes of the graph of f ( x ) ? We separate out the terms: lim x 2 - x 2 x 2 - 4 = lim x 2 - x 2 x + 2 1 x - 2 .
Image of page 5

Subscribe to view the full document.

Math 1110 Prelim I (9/27/2011) 6 The limit lim x 2 - x 2 x + 2 = 1 . On the other hand, lim x 2 - 1 x - 2 = - because the denominator will be a very small negative number, so its reciprocal will be a very large positive number.
Image of page 6
You've reached the end of this preview.
  • Fall '06
  • MARTIN,C.
  • lim, Limit of a function

{[ 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