Unit 1

Let f be a function such that limx→5−f(x)=∞. Which of the following statements must be true?

C: The graph of f has a vertical asymptote at x=5.

The table above gives selected values for a function f. Based on the data in the table, which of the following could not be the graph of f on the interval 1.9≤x≤2.1 ?

D

Let f be the function defined above. For what value of c, if any, is f continuous at x=3 ?

D: There is no such c.

The table above gives selected values for a continuous function f. Based on the data in the table, which of the following is the best approximation for limx→3f(x) ?

C: 5

Let f be a function such that f(5)<6<f(7). Which of the following statements provides sufficient additional information to conclude that there is a value x=c in the interval [5,7] such that f(c)=6 ?

C: F is continuous for all x

The function f is given by f(x)=0.1x4−0.5x3−3.3x2+7.7x−1.99. For how many positive values of b does limx→bf(x)=2 ?

C: Thre

Let f be the function given by f(x)=x+tan(x5)−10. The Intermediate Value Theorem applied to f on the closed interval [12,15] guarantees a solution in [12,15] to which of the following equations?

C: f(x)=4

Which of the following functions is continuous at x=3 ?

C: h(x)=−8sin(π2x)8−8cos(πx)forx<3forx=3forx>3

Let f be the function given by f(x)=e2x−1x. Which of the following equations expresses the property that f(x) can be made arbitrarily close to 2 by taking x sufficiently close to 0, but not equal to 0 ?

C: limx→0f(x)=2

The function h is defined by h(x)=x2−7/x−3. Which of the following statements must be true?

C: limx→3−h(x)=−∞limx→3−h(x)=−∞ and limx→3+h(x)=+∞limx→3+h(x)=+∞

The table above gives values of a function f at selected values of x. Which of the following conclusions is supported by the data in the table?

C: limx→4+f(x)=6

The function f has the property that as x gets closer and closer to 4, the values of f(x) get closer and closer to 7. Which of the following statements must be true?

C: limx→4f(x)=7

A particle is moving on the x-axis and the position of the particle at time t is given by x(t), whose graph is given above. Which of the following is the best estimate for the speed of the particle at time t=8 ?

A: 0

The table above gives selected values for a function f. Also shown is a portion of the graph of f. The graph consists of a line segment for x<3 and part of a parabola for x>3. What is limx→3f(x) ?

A: 1.6

If f is the function defined by f(x)=1x−1x−1, then limx→1f(x) is equivalent to which of the following?

A: limx→1(−1x)

If f(x)=sinx−1/cos2x, then limx→π/2f(x) is equivalent to which of the following?

A: limx→π/2−1/1+sinx

Let f be the function given by f(x)=∣∣x2−3∣∣⋅(x+0.5)(x2−3)(x+0.5). On which of the following open intervals is f continuous?

C: (0,1)

The function f is continuous on the interval −1<x<3 and is not continuous on the interval −1≤x≤3. Which of the following could not be an expression for f(x) ?

C: (x+1)(x-3)

Let f be the function defined by f(x)=3x20/4ex+8x20 for x>0. Which of the following is a horizontal asymptote to the graph of f ?

A: y=0

Let f be a function of x. If limx→2−f(x)=+∞ and limx→2+f(x)=−∞, which of the following could be a graph of f ?

B

Let g be the function defined above, where k is a constant. For what value of k is g continuous at x=−3 ?

B: -3/2

If f is the function defined above, then limx→1−f(x) is

B: 4

The function g is given by g(x)=7x−26x−5. The function h is given by h(x)=3x+142x+1. If f is a function that satisfies g(x)≤f(x)≤h(x) for 0<x<5, what is limx→2f(x) ?

B: 4

The graph of a function f is shown in the figure above. At what value of x does f have a removable discontinuity?

B: X=3

The function f is defined above. Which of the following statements is true?

B: f has a removable discontinuity at x=2

Let f be the function defined above. Which of the following statements is true?

B: f is not continuous at x=1 because f(1) does not exist.

The graph of the function f is shown above. Which of the following expressions equals 2 ?

B: limx→3−f(x)

If f is the function defined by f(x)=x−9/x√−3, then limx→9f(x) is equivalent to which of the following?

B: limx→9(x√+3)

A function f satisfies limx→1f(x)=3. Which of the following could be the graph of f?

C

The function f has a jump discontinuity at x=3. Which of the following could be the graph of f ?

C

Let f be a function of x. Which of the following statements, if true, would guarantee that there is a number c in the interval [−2,3] such that f(c)=10 ?

C. f is continuous on the interval [−2,3][−2,3], where f(−2)=0f(−2)=0 and f(3)=20f(3)=20.

The population on an island is modeled by P(t)=6000/40+60e−0.03t for t≥0, where P(t) is the number of people on the island after t years. What is limt→∞P(t) ?

C: 150

The graphs of the functions f and g are shown above. The value of limx→4f(x)+7g(x) is

C: 2

limx→0cosx+3ex/2ex is

C: 2

Let f and g be functions such that limx→4g(x)=2 and limx→4f(x)g(x)=π. What is limx→4f(x) ?

C: 2pie

limx→07x5+5x2+12x/3x5+4x is

C: 3

Let f be a function of x. The value of limx→af(x) can be found using the squeeze theorem with the functions g and h. Which of the following could be graphs of f, g, and h ?

D

The graph of the function f is shown above. On which of the following intervals is f continuous?

D: (3,5)

Let f be the function defined by f(x)=2x+3/x+1. Which of the following statements are true?

D: 1, 2, and 3

Let f be the piecewise function defined above. Also shown is a portion of the graph of f. What is the value of limx→2f(f(x)) ?

D: 1/2

If limx→6f(x) exists with limx→6f(x)<5 and f(6)=10, which of the following statements must be false?

D: F is continuous at x=6

The position of a particle moving to the right on the x-axis is given by x(t), where x(t) is measured in inches and t is measured in minutes for 0≤t≤100. If y=x(t) is a linear function, which of the following would most likely give the best estimate of the speed of the particle, in inches per minute, at time t=20 minutes?

D: The slope of the graph of y=x(t)

The function f is defined for all x in the interval 4<x<6. Which of the following statements, if true, implies that limx→5f(x)=17 ?

D: There exist functions g and hh with g(x)≤f(x)≤h(x)g(x)≤f(x)≤h(x) for 4<x<64<x<6, and limx→5g(x)=limx→5h(x)=17limx→5g(x)=limx→5h(x)=17.

The table above gives values of the function f at selected values of x. Which of the following statements must be true?

D: limx→1f(x) cannot be definitively determined from the data in the table.

If f is the function defined above, then limx→0f(x) is

D: nonexistent

The graph of the function f is shown above. The value of limx→5f(x) is

D: nonexistent

A rocket leaves the surface of Earth at time t=0 and travels straight up from the surface. The height, in feet, of the rocket above the surface of Earth is given by y(t), where t is measured in seconds for 0≤t≤600. Values of y(t) for selected values of t are given in the table above. Of the following values of t, at which value would the speed of the rocket most likely be greatest based on the data in the table?

D: t=400

Let f be the function defined above. For what values of b is f continuous at x=2 ?

D: −2.998 and 0.647