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Untitled Page 19

  • Page ID
    148358
  • Homework Problems

    1. (a) Prove, using the Weierstrass definition of the limit, that if \lim_{x→ a} f(x) = F and \lim_{x→ a} g(x) = G both exist, them \lim_{x→ a} [f(x)+g(x)] = F+G, i.e., that the limit of a sum is the sum of the limits. (b) Prove the same thing using the definition of the limit in terms of infinitesimals. (solution in the pdf version of the book)

    2. Sketch the graph of the function e-1/x, and evaluate the following four limits:

    eq_6697ecc7.png

    x}

    x}

    x}

    (solution in the pdf version of the book)

    3. Verify the following limits.

    eq_61dcd2ff.png

    eq_c5fac93e.png

    eq_337d404f.png

    eq_4a796f94.png

    eq_5d35e1a2.png

    [Granville, 1911] (solution in the pdf version of the book)

    4. Evaluate

    eq_08274d81.png

    exactly, and check your result by numerical approximation. (solution in the pdf version of the book)

    5. Amy is asked to evaluate

    eq_898002d0.png

    She applies l'H\^{o}pital's rule, differentiating top and bottom to find 1/ex, which equals 1 when she plugs in x=0. What is wrong with her reasoning? (solution in the pdf version of the book)

    6. Evaluate

    eq_c66f5932.png

    exactly, and check your result by numerical approximation. (solution in the pdf version of the book)

    7. Evaluate

    eq_4116484b.png

    exactly, and check your result by numerical approximation. (solution in the pdf version of the book)

    8. Prove a form of l'H\^{o}pital's rule stating that

    eq_ab6273c9.png

    is equal to the limit of f'/g' at infinity. Hint: change to some new variable u such that x→∞ corresponds to u→0. (solution in the pdf version of the book)

    9. Prove that the linear function y=ax+b, where a and b are real, is continuous, first using the definition of continuity in terms of infinitesimals, and then using the definition in terms of the Weierstrass limit. (solution in the pdf version of the book)

    (c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.