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AP®MathsCollege BoardCalculus BCExam QuestionsUnit 1: Limits & ContinuityLimits

Limits (College Board AP® Calculus BC): Exam Questions

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1
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The function f is continuous at all points except x equals 0. The table below gives the values of f for a number of different values of x.

x

f(x)

-0.1

0.99833417

-0.01

0.99998333

-0.001

0.99999983

0

not defined

0.001

0.99999983

0.01

0.99998333

0.1

0.99833417

What does the table suggest that the value of limit as x rightwards arrow 0 of f open parentheses x close parentheses is?

  • undefined

  • 0

  • 1

  • infinity

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2
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Graph of function 'f' with a filled point at (1,-2) and open points at (1,1) and (2,-3), illustrating the curve's changing behavior on a coordinate plane.

The graph of the function f is shown in the figure above. The value of limit as x rightwards arrow 1 to the power of plus of f open parentheses x close parentheses is

  • negative 3

  • negative 2

  • 1

  • nonexistent

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3
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Let f and g be functions such that limit as x rightwards arrow 2 of f open parentheses x close parentheses equals 4 and limit as x rightwards arrow 2 of g open parentheses x close parentheses equals negative 3.

If the function h is defined by h open parentheses x close parentheses equals 3 f open parentheses x close parentheses plus g open parentheses x close parentheses, then limit as x rightwards arrow 2 of h open parentheses x close parentheses is

  • undefined

  • -5

  • 1

  • 9

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4
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The line x equals negative 2 is a vertical asymptote to the graph of which of the following functions?

  • y equals fraction numerator 1 over denominator x plus 2 end fraction

  • y equals x plus 2

  • y equals negative 2 x

  • y equals e to the power of x plus 2 end exponent

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5
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limit as x rightwards arrow infinity of open parentheses 4 minus 3 over x close parentheses is

  • nonexistent

  • 0

  • 1

  • 4

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6
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Graph with two line segments. The first connects (-4, 2) to (1, 1) with an open circle at (1, 1). The second connects (1, 2) to (4, 1).

The figure above shows the graph of a function f with domain negative 4 less or equal than x less or equal than 4 Which of the following statements are true?

I. limit as x rightwards arrow 1 to the power of minus of f open parentheses x close parentheses exists

II. limit as x rightwards arrow 1 to the power of plus of f open parentheses x close parentheses exists

III. limit as x rightwards arrow 1 of f open parentheses x close parentheses exists

  • I only

  • II only

  • I and II only

  • I, II and III

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7
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limit as x rightwards arrow infinity of fraction numerator 2 x to the power of 5 plus 4 x squared plus x minus 3 over denominator 5 x to the power of 5 plus x cubed minus 2 x squared plus 4 end fraction equals

  • negative 3 over 4

  • 0

  • 2 over 5

  • nonexistent

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1
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A graph labeled "Graph of f" plotting points (1,3), (2,1), (3,3), (4,5), (5,3), and (6,5) on a grid, with x-axis from 0 to 6, y-axis from 0 to 6.

The graph of the function f is shown above. Which of the following statements is false?

  • limit as x rightwards arrow 2 of f open parentheses x close parentheses exists.

  • limit as x rightwards arrow 3 of f open parentheses x close parentheses exists.

  • limit as x rightwards arrow 5 of f open parentheses x close parentheses exists.

  • The function is continuous at x equals 4.

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2
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The line y equals 3 is a horizontal asymptote to the graph of which of the following functions?

  • y equals fraction numerator 1 over denominator x minus 3 end fraction

  • y equals 3 x

  • y equals fraction numerator x minus 3 x squared over denominator 1 minus x squared end fraction

  • y equals fraction numerator 3 x over denominator 7 minus x end fraction

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3
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If f open parentheses x close parentheses equals open curly brackets table row cell ln x space space space space space space for space 0 less than x less or equal than 3 end cell row cell x ln 3 space space space space for space 3 less than x less or equal than 5 end cell end table close, then limit as x rightwards arrow 3 of f open parentheses x close parentheses is

  • ln 3

  • ln 27

  • 1 third

  • nonexistent

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4
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Let f be a function such that limit as x rightwards arrow 0 of f open parentheses x close parentheses equals negative 1.

Let g be the function defined by g open parentheses x close parentheses equals sin x.

What is limit as x rightwards arrow 0 to the power of minus of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction?

  • negative infinity

  • -1

  • 0

  • plus infinity

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5
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Let f, g and h be functions defined on the interval 1 less than x less than 3. limit as x rightwards arrow 2 of g open parentheses x close parentheses equals limit as x rightwards arrow 2 of h open parentheses x close parentheses equals negative 5, and it is also true that g open parentheses x close parentheses less or equal than f open parentheses x close parentheses less or equal than h open parentheses x close parentheses for all x such that 1 less than x less than 3.

Which one of the following results allows you to determine that limit as x rightwards arrow 2 of f open parentheses x close parentheses equals negative 5?

  • fundamental theorem of calculus

  • intermediate value theorem

  • mean value theorem

  • squeeze theorem

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6
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limit as x rightwards arrow infinity of fraction numerator open parentheses 3 x minus 2 close parentheses open parentheses 1 minus x close parentheses over denominator open parentheses x minus 2 close parentheses open parentheses x plus 1 close parentheses end fraction is

  • -3

  • 1

  • 3

  • nonexistent

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7
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limit as x rightwards arrow 0 of fraction numerator 7 x to the power of 5 minus 3 x to the power of 4 over denominator 9 x to the power of 5 plus 2 x to the power of 4 end fraction is

  • negative 3 over 2

  • 0

  • 7 over 9

  • nonexistent

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8
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The domain of a function f is x less or equal than 0. The horizontal line y equals negative 1 is an asymptote for the graph of f. Which of the following statements must be true?

  • limit as x rightwards arrow negative 1 of f open parentheses x close parentheses equals infinity

  • limit as x rightwards arrow 0 of f open parentheses x close parentheses equals negative 1

  • limit as x rightwards arrow negative infinity of f open parentheses x close parentheses equals negative 1

  • f open parentheses x close parentheses not equal to negative 1 for all x less or equal than 0

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9
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For which of the following does limit as x rightwards arrow 2 of f open parentheses x close parentheses exist?

Three graphs labelled I, II, III show functions with open and closed dots at various coordinates, each titled "Graph of f" on x-y axes. I has two disjointed curves. II and III both have one curve with a hole at x=2. II has a closed value at x=2 below the hole, whereas III does not have a closed value at x=2.

  • I only

  • II only

  • I and III only

  • II and III only

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1
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If limit as x rightwards arrow 2 of f open parentheses x close parentheses equals 5, which of the following must be true?

I. space space f is continuous at x equals 2.

II. space space f is differentiable at x equals 2.

III. space space f open parentheses 2 close parentheses equals 5

  • None

  • I only

  • I and III only

  • I, II and III

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2
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Let f, g and h be continuous functions such that limit as x rightwards arrow 5 of f open parentheses x close parentheses equals negative 2, limit as x rightwards arrow 3 of g open parentheses x close parentheses equals 5 and limit as x rightwards arrow 3 of h open parentheses x close parentheses equals negative 7.

If the function p is defined by p open parentheses x close parentheses equals fraction numerator f open parentheses g open parentheses x close parentheses close parentheses over denominator open square brackets h open parentheses x close parentheses close square brackets squared end fraction, then limit as x rightwards arrow 3 of p open parentheses x close parentheses is

  • negative 2 over 49

  • 5 over 49

  • 2 over 7

  • unable to be determined from the information provided

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3
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limit as x rightwards arrow 2 of open parentheses fraction numerator square root of x plus 1 end root minus square root of 3 over denominator x minus 2 end fraction close parentheses equals

  • 0 over 0

  • fraction numerator square root of 3 over denominator 6 end fraction

  • square root of 6

  • 2 square root of 3

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4
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Let f and g be the functions defined by f open parentheses x close parentheses equals x squared minus 2 x plus 2 and g open parentheses x close parentheses equals sin open parentheses fraction numerator pi x over denominator 2 end fraction close parentheses. Let h be a function defined for all real numbers and satisfying f open parentheses x close parentheses less or equal than h open parentheses x close parentheses less or equal than g open parentheses x close parentheses for all values of x except x equals 1.

What is limit as x rightwards arrow 1 of h open parentheses x close parentheses?

  • -1

  • 0

  • 1

  • unable to be determined from the information provided

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5
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If

f open parentheses x close parentheses equals open curly brackets table row cell fraction numerator sin open parentheses 7 minus 7 x close parentheses over denominator sin open parentheses 2 x minus 2 close parentheses end fraction space space space space space space for space minus 1 half less or equal than x less than 1 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space for space x equals 1 end cell row cell fraction numerator x squared minus 3 x plus 9 over denominator 2 end fraction space space space space space space for space x greater than 1 end cell end table close

then limit as x rightwards arrow 1 of f open parentheses x close parentheses is

  • negative 7 over 2

  • 0

  • 7 over 2

  • nonexistent

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6
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Graph of function f with points at (-3,4) and (0, 2). The function has holes at (-3, 2), (-2,2) and (3,2).

The graph of a function f is shown above. For which of the following values of c does limit as x rightwards arrow c of f open parentheses x close parentheses equals 2?

  • 0 only

  • -3 and 0 only

  • -3, 0, and 3 only

  • -3, -2, 0 and 3

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7
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What are all the horizontal asymptotes of the graph of y equals fraction numerator 3 plus 5 to the power of x over denominator 2 plus 5 to the power of x end fraction in the x y-plane.

  • y equals 1 only

  • y equals 3 over 2 only

  • y equals 1 and y equals 3 over 2 only

  • y equals 0 and y equals 3 over 2 only

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