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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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A chemical is added to the water in a swimming pool. The amount, in grams per liter of water, of the chemical in the water at time t hours is modeled by a function A open parentheses t close parentheses.

Using correct units, interpret the statement limit as t rightwards arrow infinity of A open parentheses t close parentheses equals 3.7 in the context of this problem.

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2
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Functions f and g are continuous functions with f open parentheses x close parentheses less or equal than g open parentheses x close parentheses for 0 less than x less than 5 and f open parentheses 3 close parentheses equals g open parentheses 3 close parentheses equals 1.

Let h be a function 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 0 less than x less than 5. Is h continuous at x equals 3? Justify your answer.

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3
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Let f be the function given by f open parentheses x close parentheses equals sin open parentheses pi over 2 minus 1 over x close parentheses for all x greater than 0.

Find limit as x rightwards arrow infinity of f open parentheses x close parentheses.

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4a
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Let f be the function given by f open parentheses x close parentheses equals fraction numerator tan open parentheses pi x close parentheses over denominator ln open parentheses 2 x close parentheses end fraction for all 0 less than x less than 1 half.

Find limit as x rightwards arrow 0 to the power of plus of f open parentheses x close parentheses.

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4b
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Find limit as x rightwards arrow 1 half to the power of minus of f open parentheses x close parentheses.

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5a
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Graph of function f which is made up of two curves is drawn on a grid, with an open point at (2,3) and a closed point at (2,1).

The figure above shows the graph of a function f.

Find limit as x rightwards arrow 2 to the power of minus of f open parentheses x close parentheses and limit as x rightwards arrow 2 to the power of plus of f open parentheses x close parentheses.

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5b
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Find limit as x rightwards arrow 2 of f open parentheses x close parentheses or explain why it does not exist.

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6
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Find limit as x rightwards arrow negative 1 of fraction numerator x squared plus 6 x plus 5 over denominator x squared minus 2 x minus 3 end fraction.

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1
Sme Calculator2 marks

Particle P moves along the x-axis such that, for time t greater than 0, its position is given by x subscript P open parentheses t close parentheses equals 2 plus 3 e to the power of negative t end exponent.

Particle Q moves along the y-axis such that, for time t greater than 0, its position is given by y subscript Q open parentheses t close parentheses equals 1 over t minus 3.

As t rightwards arrow infinity, which particle will eventually be farther from the origin? Give a reason for your answer.

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2
Sme Calculator1 mark
A graph of the function f described inn the question, consisting of line segments between (-2, 1) and (3, 0), and between (3, 0) and (3, -3), and a quarter circle connecting (3, -3) to (6, 0)

The continuous function f is defined on the closed interval negative 2 less or equal than x less or equal than 6. The figure above shows the graph of f, consisting of two line segments and a quarter of a circle centered at the point open parentheses 6 comma space minus 3 close parentheses.

Find limit as x rightwards arrow 1 of open parentheses fraction numerator 5 to the power of x minus 2 f open parentheses x close parentheses over denominator arctan x plus 3 f to the power of apostrophe open parentheses x close parentheses end fraction close parentheses.

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3
Sme Calculator2 marks

Let f and g be the functions defined by f open parentheses x close parentheses equals 3 plus cos x and g open parentheses x close parentheses equals 2 minus open parentheses x minus pi close parentheses squared. It is known that g open parentheses x close parentheses less or equal than f open parentheses x close parentheses for 0 less than x less than 2 pi.

Let h be a function such that g open parentheses x close parentheses less or equal than h open parentheses x close parentheses less or equal than f open parentheses x close parentheses for 0 less than x less than 2 pi.

Find limit as x rightwards arrow pi of h open parentheses x close parentheses, being sure to justify your answer.

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4
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Find limit as x rightwards arrow infinity of fraction numerator 2 x to the power of 15 minus 3 over denominator 5 x to the power of 15 plus 7 end fraction or show that it does not exist.

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5
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Find limit as x rightwards arrow 3 of fraction numerator 2 x squared minus x minus 15 over denominator x squared minus 6 x plus 9 end fraction or show that it does not exist.

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1
Sme Calculator2 marks

Find limit as x rightwards arrow 0 of open parentheses fraction numerator 3 x over denominator square root of 2 minus square root of 2 minus x end root end fraction close parentheses.

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2a
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It is known that cos x less or equal than fraction numerator sin x over denominator x end fraction less or equal than fraction numerator 1 over denominator cos x end fraction for negative pi over 2 less than x less than 0 and 0 less than x less than pi over 2.

Find limit as x rightwards arrow 0 of fraction numerator sin x over denominator x end fraction. Justify your answer.

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2b
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Find limit as x rightwards arrow 0 of fraction numerator sin x cos 2 x over denominator 3 x end fraction.

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3
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Graph showing a V-shaped function with lines intersecting at the origin; x and y axes range from -4 to 4. Title: Graph of f.

The continuous function f is defined on the closed interval negative 4 less or equal than x less or equal than 4. The figure above shows the graph of f, consisting of two line segments which intersect at open parentheses 0 comma space 0 close parentheses. Let g be the function defined by g open parentheses x close parentheses equals f to the power of apostrophe open parentheses x close parentheses.

Find limit as x rightwards arrow 0 of g open parentheses x close parentheses or show that it does not exist.

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4
Sme Calculator2 marks

Find limit as x rightwards arrow pi of fraction numerator square root of e to the power of sin x end exponent plus 2 end root minus square root of 3 over denominator e to the power of sin x end exponent cos x minus cos x end fraction.

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5
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Find limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus 5 x end root minus 2 x close parentheses or show that it does not exist.

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