Understanding Limits: Get Stronger

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. Complete the following table with the appropriate values: [latex]y[/latex]-coordinate of Q, the point [latex]Q(x,y),[/latex] and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits.
    [latex]x[/latex] [latex]y[/latex] [latex]Q(x,y)[/latex] [latex]m_{\sec}[/latex]
    [latex]1.1[/latex] a. e. i.
    [latex]1.01[/latex] b. f. j.
    [latex]1.001[/latex] c. g. k.
    [latex]1.0001[/latex] d. h. l.
  2. Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to [latex]f[/latex] at [latex]x=1[/latex].
  3. Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[/latex]. Graph [latex]f(x)[/latex] and the tangent line.

For the following exercises (4-6), points [latex]P(1,1)[/latex] and [latex]Q(x,y)[/latex] are on the graph of the function [latex]f(x)=x^3[/latex].

  1. Complete the following table with the appropriate values: [latex]y[/latex]-coordinate of [latex]Q[/latex], the point [latex]Q(x,y)[/latex], and the slope of the secant line passing through points [latex]P[/latex] and [latex]Q[/latex]. Round your answer to eight significant digits.
    [latex]x[/latex] [latex]y[/latex] [latex]Q(x,y)[/latex] [latex]m_{\sec}[/latex]
    [latex]1.1[/latex] a. e. i.
    [latex]1.01[/latex] b. f. j.
    [latex]1.001[/latex] c. g. k.
    [latex]1.0001[/latex] d. h. l.
  2. Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to [latex]f[/latex] at [latex]x=1[/latex].
  3. Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[/latex]. Graph [latex]f(x)[/latex] and the tangent line.

For the following exercises (7-9), points [latex]P(4,2)[/latex] and [latex]Q(x,y)[/latex] are on the graph of the function [latex]f(x)=\sqrt{x}[/latex].

  1. Complete the following table with the appropriate values: [latex]y[/latex]-coordinate of [latex]Q[/latex], the point [latex]Q(x,y)[/latex], and the slope of the secant line passing through points [latex]P[/latex] and [latex]Q[/latex]. Round your answer to eight significant digits.
    [latex]x[/latex] [latex]y[/latex] [latex]Q(x,y)[/latex] [latex]m_{\sec}[/latex]
    [latex]4.1[/latex] a. e. i.
    [latex]4.01[/latex] b. f. j.
    [latex]4.001[/latex] c. g. k.
    [latex]4.0001[/latex] d. h. l.
  2. Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to [latex]f[/latex] at [latex]x=4[/latex].
  3. Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[/latex].

For the following exercises (10-12), points [latex]P(1.5,0)[/latex] and [latex]Q(\phi ,y)[/latex] are on the graph of the function [latex]f(\phi )= \cos (\pi \phi )[/latex].

  1. Complete the following table with the appropriate values: [latex]y[/latex]-coordinate of [latex]Q[/latex], the point [latex]Q(x,y)[/latex], and the slope of the secant line passing through points [latex]P[/latex] and [latex]Q[/latex]. Round your answer to eight significant digits.
    [latex]x[/latex] [latex]y[/latex] [latex]Q(\phi ,y)[/latex] [latex]m_{\sec}[/latex]
    [latex]1.4[/latex] a. e. i.
    [latex]1.49[/latex] b. f. j.
    [latex]1.499[/latex] c. g. k.
    [latex]1.4999[/latex] d. h. l.
  2. Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to [latex]f[/latex] at [latex]x=4[/latex].
  3. Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[/latex].

For the following exercises (13-15), points [latex]P(-1,-1)[/latex] and [latex]Q(x,y)[/latex] are on the graph of the function [latex]f(x)=\dfrac{1}{x}[/latex].

  1. Complete the following table with the appropriate values: [latex]y[/latex]-coordinate of [latex]Q[/latex], the point [latex]Q(x,y)[/latex], and the slope of the secant line passing through points [latex]P[/latex] and [latex]Q[/latex]. Round your answer to eight significant digits.
    [latex]x[/latex] [latex]y[/latex] [latex]Q(x,y)[/latex] [latex]m_{\sec}[/latex]
    [latex]-1.05[/latex] a. e. i.
    [latex]-1.01[/latex] b. f. j.
    −[latex]-1.005[/latex] c. g. k.
    [latex]-1.001[/latex] d. h. l.
  2. Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to [latex]f[/latex] at [latex]x=-1[/latex].
  3. Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[/latex].

For the following exercises (16-17), the position function of a ball dropped from the top of a 200-meter tall building is given by [latex]s(t)=200-4.9t^2[/latex], where position [latex]s[/latex] is measured in meters and time [latex]t[/latex] is measured in seconds. Round your answer to eight significant digits.

  1. Compute the average velocity of the ball over the given time intervals.
    1. [latex][4.99,5][/latex]
    2. [latex][5,5.01][/latex]
    3. [latex][4.999,5][/latex]
    4. [latex][5,5.001][/latex]
  2. Use the preceding exercise to guess the instantaneous velocity of the ball at [latex]t=5[/latex] sec.

For the following exercises (18-19), consider a stone tossed into the air from ground level with an initial velocity of 15 m/sec. Its height in meters at time [latex]t[/latex] seconds is [latex]h(t)=15t-4.9t^2[/latex].

  1. Compute the average velocity of the stone over the given time intervals.
    1. [latex][1,1.05][/latex]
    2. [latex][1,1.01][/latex]
    3. [latex][1,1.005][/latex]
    4. [latex][1,1.001][/latex]
  2. Use the preceding exercise to guess the instantaneous velocity of the stone at [latex]t=1[/latex] sec.

For the following exercises (20-21), consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by [latex]h(t)=600+78.4t-4.9t^2[/latex], where [latex]t[/latex] is measured in seconds.

  1. Compute the average velocity of the rocket over the given time intervals.
    1. [latex][9,9.01][/latex]
    2. [latex][8.99,9][/latex]
    3. [latex][9,9.001][/latex]
    4. [latex][8.999,9][/latex]
  2. Use the preceding exercise to guess the instantaneous velocity of the rocket at [latex]t=9[/latex] sec.

For the following exercises (22-23), consider an athlete running a 40-m dash. The position of the athlete is given by [latex]d(t)=\dfrac{t^3}{6}+4t[/latex], where [latex]d[/latex] is the position in meters and [latex]t[/latex] is the time elapsed, measured in seconds.

  1. Compute the average velocity of the runner over the given time intervals.
    1. [latex][1.95,2.05][/latex]
    2. [latex][1.995,2.005][/latex]
    3. [latex][1.9995,2.0005][/latex]
    4. [latex][2,2.00001][/latex]
  2. Use the preceding exercise to guess the instantaneous velocity of the runner at [latex]t=2[/latex] sec.

For the following exercises (24-25), consider the function [latex]f(x)=|x|[/latex].

  1. Sketch the graph of [latex]f[/latex] over the interval [latex][-1,2][/latex] and shade the region above the [latex]x[/latex]-axis.
  2. Use the preceding exercise to find the exact value of the area between the [latex]x[/latex]-axis and the graph of [latex]f[/latex] over the interval [latex][-1,2][/latex] using rectangles. For the rectangles, use the square units, and approximate both above and below the lines. Use geometry to find the exact answer.

For the following exercises (26-27), consider the function [latex]f(x)=\sqrt{1-x^2}[/latex]. (Hint: This is the upper half of a circle of radius [latex]1[/latex] positioned at [latex](0,0)[/latex].)

  1. Sketch the graph of [latex]f[/latex] over the interval [latex][-1,1][/latex].
  2. Use the preceding exercise to find the exact area between the [latex]x[/latex]-axis and the graph of [latex]f[/latex] over the interval [latex][-1,1][/latex] using rectangles. For the rectangles, use squares [/latex]0.4[/latex] by [latex]0.4[/latex] units, and approximate both above and below the lines. Use geometry to find the exact answer.

For the following exercises (28-29), consider the function [latex]f(x)=−x^2+1[/latex].

  1. Sketch the graph of [latex]f[/latex] over the interval [latex][-1,1][/latex].
  2. Approximate the area of the region between the [latex]x[/latex]-axis and the graph of [latex]f[/latex] over the interval [latex][-1,1][/latex].

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]\underset{x\to 1}{\lim}(1-2x)[/latex]
    [latex]x[/latex] [latex]1-2x[/latex] [latex]x[/latex] [latex]1-2x[/latex]
    0.9 a. [latex]1.1[/latex] e.
    0.99 b. [latex]1.01[/latex] f.
    0.999 c. [latex]1.001[/latex] g.
    0.9999 d. [latex]1.0001[/latex] h.
  2. [latex]\underset{z\to 0}{\lim}\dfrac{z-1}{z^2(z+3)}[/latex]
    [latex]z[/latex] [latex]\frac{z-1}{z^2(z+3)}[/latex] [latex]z[/latex] [latex]\frac{z-1}{z^2(z+3)}[/latex]
    [latex]−0.1[/latex] a. [latex]0.1[/latex] e.
    [latex]−0.01[/latex] b. [latex]0.01[/latex] f.
    [latex]−0.001[/latex] c. [latex]0.001[/latex] g.
    [latex]−0.0001[/latex] d. [latex]0.0001[/latex] h.
  3. [latex]\underset{x\to 2^-}{\lim}\dfrac{1-\frac{2}{x}}{x^2-4}[/latex]
    [latex]x[/latex] [latex]\frac{1-\frac{2}{x}}{x^2-4}[/latex] [latex]x[/latex] [latex]\frac{1-\frac{2}{x}}{x^2-4}[/latex]
    [latex]1.9[/latex] a. [latex]2.1[/latex] e.
    [latex]1.99[/latex] b. [latex]2.01[/latex] f.
    [latex]1.999[/latex] c. [latex]2.001[/latex] g.
    [latex]1.9999[/latex] d. [latex]2.0001[/latex] h.

In the following exercise, set up a table of values and round to eight significant digits. Based on the table of values, make a guess about what the limit is. Then, use a calculator to graph the function and determine the limit. Was the conjecture correct? If not, why does the method of tables fail?

  1. [latex]\underset{\alpha \to 0^+}{\lim}\frac{1}{\alpha} \cos \left(\frac{\pi }{\alpha }\right)[/latex]
    [latex]a[/latex] [latex]\frac{1}{\alpha } \cos (\frac{\pi }{\alpha })[/latex]
    [latex]0.1[/latex] a.
    [latex]0.01[/latex] b.
    [latex]0.001[/latex] c.
    [latex]0.0001[/latex] d.

In the following exercises (5-6), consider the graph of the function [latex]y=f(x)[/latex] shown here. Which of the statements about [latex]y=f(x)[/latex] are true and which are false? Explain why a statement is false.

A graph of a piecewise function with three segments and a point. The first segment is a curve opening upward with vertex at (-8, -6). This vertex is an open circle, and there is a closed circle instead at (-8, -3). The segment ends at (-2,3), where there is a closed circle. The second segment stretches up asymptotically to infinity along x=-2, changes direction to increasing at about (0,1.25), increases until about (2.25, 3), and decreases until (6,2), where there is an open circle. The last segment starts at (6,5), increases slightly, and then decreases into quadrant four, crossing the x axis at (10,0). All of the changes in direction are smooth curves.

  1. [latex]\underset{x\to -2^+}{\lim}f(x)=3[/latex]
     
  2. [latex]\underset{x\to 6}{\lim}f(x)=5[/latex]

In the following exercises (7-9), use the following graph of the function [latex]y=f(x)[/latex] to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with two segments. The first segment exists for x <=1, and the second segment exists for x > 1. The first segment is linear with a slope of 1 and goes through the origin. Its endpoint is a closed circle at (1,1). The second segment is also linear with a slope of -1. It begins with the open circle at (1,2).

  1. [latex]\underset{x\to 1^-}{\lim}f(x)[/latex]
  2. [latex]\underset{x\to 1^+}{\lim}f(x)[/latex]
  3. [latex]\underset{x\to 1}{\lim}f(x)[/latex]

In the following exercises 10-11), use the graph of the function [latex]y=f(x)[/latex] shown here to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with two segments. The first is a linear function for x < 0. There is an open circle at (0,1), and its slope is -1. The second segment is the right half of a parabola opening upward. Its vertex is a closed circle at (0, -4), and it goes through the point (2,0).

  1. [latex]\underset{x\to 0^-}{\lim}f(x)[/latex]
  2. [latex]\underset{x\to 0}{\lim}f(x)[/latex]

In the following exercises (12-17), use the graph of the function [latex]y=f(x)[/latex] shown here to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with three segments, all linear. The first exists for x < -2, has a slope of 1, and ends at the open circle at (-2, 0). The second exists over the interval [-2, 2], has a slope of -1, goes through the origin, and has closed circles at its endpoints (-2, 2) and (2,-2). The third exists for x>2, has a slope of 1, and begins at the open circle (2,2).

  1. [latex]\underset{x\to -2^-}{\lim}f(x)[/latex]
  2. [latex]\underset{x\to -2^+}{\lim}f(x)[/latex]
  3. [latex]\underset{x\to -2}{\lim}f(x)[/latex]
  4. [latex]\underset{x\to 2^-}{\lim}f(x)[/latex]
  5. [latex]\underset{x\to 2^+}{\lim}f(x)[/latex]
  6. [latex]\underset{x\to 2}{\lim}f(x)[/latex]

In the following exercises (18-20), use the graph of the function [latex]y=g(x)[/latex] shown here to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with two segments. The first exists for x>=0 and is the left half of an upward opening parabola with vertex at the closed circle (0,3). The second exists for x>0 and is the right half of a downward opening parabola with vertex at the open circle (0,0).

  1. [latex]\underset{x\to 0^-}{\lim}g(x)[/latex]
  2. [latex]\underset{x\to 0^+}{\lim}g(x)[/latex]
  3. [latex]\underset{x\to 0}{\lim}g(x)[/latex]

In the following exercises (21-23), use the graph of the function [latex]y=h(x)[/latex] shown here to find the values, if possible. Estimate when necessary.

A graph of a function with two curves approaching 0 from quadrant 1 and quadrant 3. The curve in quadrant one appears to be the top half of a parabola opening to the right of the y axis along the x axis with vertex at the origin. The curve in quadrant three appears to be the left half of a parabola opening downward with vertex at the origin.

  1. [latex]\underset{x\to 0^-}{\lim}h(x)[/latex]
  2. [latex]\underset{x\to 0^+}{\lim}h(x)[/latex]
  3. [latex]\underset{x\to 0}{\lim}h(x)[/latex]

In the following exercises (24-25), sketch the graph of a function with the given properties.

  1. [latex]\underset{x\to -\infty }{\lim}f(x)=0, \, \underset{x\to -1^-}{\lim}f(x)=−\infty[/latex], [latex]\underset{x\to -1^+}{\lim}f(x)=\infty, \, \underset{x\to 0}{\lim}f(x)=f(0), \, f(0)=1, \, \underset{x\to \infty }{\lim}f(x)=−\infty[/latex]
  2. [latex]\underset{x\to -\infty }{\lim}f(x)=2, \, \underset{x\to -2}{\lim}f(x)=−\infty[/latex],[latex]\underset{x\to \infty }{\lim}f(x)=2, \, f(0)=0[/latex]