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AP®MathsCollege BoardCalculus ABExam QuestionsUnit 1: Limits & ContinuityLimits

Limits (College Board AP® Calculus AB)

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1
Sme Calculator1 mark

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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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
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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
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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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1
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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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