Understanding Limits: Get Stronger Answer Key

A Preview of Calculus

For the following exercises (1-3), points [latex]P(1,2)[/latex] and [latex]Q(x,y)[/latex] are on the graph of the function [latex]f(x)=x^2+1[/latex].

    1. [latex]2.2100000[/latex]
    2. [latex]2.0201000[/latex]
    3. [latex]2.0020010[/latex]
    4. [latex]2.0002000[/latex]
    5. [latex](1.1000000, 2.2100000)[/latex]
    6. [latex](1.0100000, 2.0201000)[/latex]
    7. [latex](1.0010000, 2.0020010)[/latex]
    8. [latex](1.0001000, 2.0002000)[/latex]
    9. [latex]2.1000000[/latex]
    10. [latex]2.0100000[/latex]
    11. [latex]2.0010000[/latex]
    12. [latex]2.0001000[/latex]
  2.  
  3. [latex]y=2x[/latex]
  4.  
  5. [latex]3[/latex]
  6.  
    1. [latex]2.0248457[/latex]
    2. [latex]2.0024984[/latex]
    3. [latex]2.0002500[/latex]
    4. [latex]2.0000250[/latex]
    5. [latex](4.1000000,2.0248457)[/latex]
    6. [latex](4.0100000,2.0024984)[/latex]
    7. [latex](4.0010000,2.0002500)[/latex]
    8. [latex](4.00010000,2.0000250)[/latex]
    9. [latex]0.24845673[/latex]
    10. [latex]0.24984395[/latex]
    11. [latex]0.24998438[/latex]
    12. [latex]0.24999844[/latex]
  8.  
  9. [latex]y=\frac{x}{4}+1[/latex]
  10.  
  11. [latex]\pi[/latex]
  12.  
    1. [latex]−0.95238095[/latex]
    2. [latex]−0.99009901[/latex]
    3. [latex]−0.99502488[/latex]
    4. [latex]−0.99900100[/latex]
    5. [latex](−1;.0500000,−0;.95238095)[/latex]
    6. [latex](−1;.0100000,−0;.9909901)[/latex]
    7. [latex](−1;.0050000,−0;.99502488)[/latex]
    8. [latex](1.0010000,−0;.99900100)[/latex]
    9. [latex]-0.95238095[/latex]
    10. [latex]−0.99009901[/latex]
    11. [latex]−0.99502488[/latex]
    12. [latex]−0.99900100 [/latex]
  14.  
  15. [latex]y=−x-2[/latex]
  16.  
  17. [latex]−49[/latex] m/sec (velocity of the ball is [latex]49[/latex] m/sec downward)
  18.  
  19. [latex]5.2[/latex] m/sec
  20.  
  21. [latex]-9.8[/latex] m/sec
  22.  
  23. [latex]6[/latex] m/sec
  24.  
  25. Under, [latex]1[/latex] unit2; over: [latex]4[/latex] unit2. The exact area of the two triangles is [latex]\frac{1}{2}(1)(1)+\frac{1}{2}(2)(2)=2.5 \text{ units}^2[/latex].
  26.  
  27. Under, [latex]0.96[/latex] unit2; over, [latex]1.92[/latex] unit2. The exact area of the semicircle with radius [latex]1[/latex] is [latex]\frac{\pi (1)^2}{2}=\frac{\pi }{2}[/latex] unit2.
  28.  
  29. Approximately [latex]1.3333333[/latex] unit2

Introduction to the limit of a function

In the following exercises (1-3), set up a table of values to find the indicated limit. Round to eight digits.

    1. [latex]−0.80000000[/latex]
    2. [latex]−0.98000000[/latex]
    3. [latex]−0.99800000[/latex]
    4. [latex]−0.99980000[/latex]
    5. [latex]−1.2000000[/latex]
    6. [latex]−1.0200000[/latex]
    7. [latex]−1.0020000[/latex]
    8. [latex]−1.0002000;[/latex]

    [latex]\underset{x\to 1}{\lim}(1-2x)=-1[/latex]

    1. [latex]−37.931934[/latex]
    2. [latex]−3377.9264[/latex]
    3. [latex]−333,777.93[/latex]
    4. [latex]−33,337,778[/latex]
    5. [latex]−29.032258[/latex]
    6. [latex]−3289.0365[/latex]
    7. [latex]−332,889.04[/latex]
    8. [latex]−33,328,889[/latex]

    [latex]\underset{x\to 0}{\lim}\frac{z-1}{z^2(z+3)}=−\infty[/latex]

    1. [latex]0.13495277[/latex]
    2. [latex]0.12594300[/latex]
    3. [latex]0.12509381[/latex]
    4. [latex]0.12500938[/latex]
    5. [latex]0.11614402[/latex]
    6. [latex]0.12406794[/latex]
    7. [latex]0.12490631[/latex]
    8. [latex]0.12499063[/latex]

    [latex]\underset{x\to 2^-}{\lim}\frac{1-\frac{2}{x}}{x^2-4}=0.1250=\frac{1}{8}[/latex]

    1. [latex]−10.00000[/latex]
    2. [latex]−100.00000[/latex]
    3. [latex]−1000.0000[/latex]
    4. [latex]−10,000.000;[/latex]

    Guess: [latex]\underset{\alpha \to 0^+}{\lim}\frac{1}{\alpha } \cos (\frac{\pi }{\alpha })=\infty[/latex], Actual: DNE

    A graph of the function (1/alpha) * cos (pi / alpha), which oscillates gently until the interval [-.2, .2], where it oscillates rapidly, going to infinity and negative infinity as it approaches the y axis.

  5. False; [latex]\underset{x\to -2^+}{\lim}f(x)=+\infty[/latex]
  6. False; [latex]\underset{x\to 6}{\lim}f(x)[/latex] DNE since [latex]\underset{x\to 6^-}{\lim}f(x)=2[/latex] and [latex]\underset{x\to 6^+}{\lim}f(x)=5[/latex].
  7.  
  8. [latex]2[/latex]
  9.  
  10. [latex]1[/latex]
  11. DNE
  12. [latex]0[/latex]
  13.  
  14. DNE
  15. [latex]2[/latex]
  16.  
  17. [latex]3[/latex]
  18.  
  19. DNE
  20.  
  21. [latex]0[/latex]
  22.  

  23. Answers may vary.

    A graph of a piecewise function with two segments. The first segment is in quadrant three and asymptotically goes to negative infinity along the y axis and 0 along the x axis. The second segment consists of two curves. The first appears to be the left half of an upward opening parabola with vertex at (0,1). The second appears to be the right half of a downward opening parabola with vertex at (0,1) as well.

  24. Answers may vary.

    A graph containing two curves. The first goes to 2 asymptotically along y=2 and to negative infinity along x = -2. The second goes to negative infinity along x=-2 and to 2 along y=2.