AP Calculus AB Introducing Limits Test with Point Values (PDF)

AP Calculus AB math test: Introducing Limits – Maths PDF

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This limits intro test for AP Calculus AB covers introducing limits (Common Core AP CALC AB 1.1, AP CALC AB 1.2, AP CALC AB 1.3, AP CALC AB 1.4), with a free PDF to print.

This test checks estimating limits from tables and graphs, one-sided limits and the cases where a limit does not exist (AP Calculus AB, 1.1, 1.2, 1.3, 1.4). It has 8 questions worth 20 points in total. Allow 45 minutes, work without notes, and show all work, since justification earns points on the AP exam.

Time and points

About this test

Grade: AP Calculus AB
Chapter: Introducing Limits
Standards: AP CALC AB 1.1, AP CALC AB 1.2, AP CALC AB 1.3, AP CALC AB 1.4
Time: 45 minutes
Total: 20 points

Limits intro test: questions

Question 1 – Estimating from a table [2 pts]

Complete a table for \(f(x)=\frac{x^{3}-8}{x-2}\) at \(x=1.9, 1.99, 2.01, 2.1\) and estimate \(\lim_{x\to 2} f(x)\).

Question 2 – Reading a graph [2 pts]

Graph of f near x = 2.

Use the graph of \(f\). Find: (a) \(\lim_{x\to 2^-} f(x)\) (b) \(\lim_{x\to 2^+} f(x)\) (c) \(f(2)\) (d) \(\lim_{x\to 2} f(x)\). (2 points)

Question 3 – Piecewise function [2 pts]

Let \(f(x)=4-x^{2}\) for \(x < -1\) and \(f(x)=2x+5\) for \(x\ge -1\). Does \(\lim_{x\to -1} f(x)\) exist? Justify.

Question 4 – Parameter for a limit [3 pts]

Let \(f(x)=kx-2\) for \(x < 3\) and \(f(x)=x^{2}-5\) for \(x\ge 3\). Find the value of \(k\) so that \(\lim_{x\to 3} f(x)\) exists, and state the limit.

Question 5 – Instantaneous speed [2 pts]

A sprinter’s distance in meters is \(s(t)=5t^{2}\) after \(t\) seconds. Find the average speed on \([2,2.1]\), \([2,2.01]\) and \([2,2.001]\), then estimate the speed at \(t=2\).

Question 6 – Unbounded behavior [3 pts]

The graph of k(x) = 1/(x − 1)².

The graph shows \(k(x)=\frac{1}{(x-1)^{2}}\). (a) Describe the outputs as \(x\to 1^-\) and as \(x\to 1^+\). (1 point) (b) Does \(\lim_{x\to 1} k(x)\) exist as a real number? Explain. (1 point) (c) Find \(k(0.9)\) and \(k(1.1)\). (1 point)

Question 7 – Taxi fare jump [3 pts]

A taxi fare in dollars is \(F(d)=8\) for \(0 < d\le 1\) miles and \(F(d)=11\) for \(1 < d\le 2\) miles. (a) Find \(\lim_{d\to 1^-} F(d)\) and \(\lim_{d\to 1^+} F(d)\). (2 points) (b) Does \(\lim_{d\to 1} F(d)\) exist? Why? (1 point)

Question 8 – Table and conclusion [3 pts]

Values of h near x = 0.

The table shows values of \(h\) near \(x=0\). (a) Estimate the left-hand and right-hand limits at \(0\). (2 points) (b) State whether \(\lim_{x\to 0} h(x)\) exists and justify. (1 point)

Key facts behind this limits intro test

Key facts

A limit describes nearby behavior, not the value at the point itself.
The two-sided limit exists only if the left-hand and right-hand limits are equal.
Tables and graphs give estimates; algebra gives exact values.
A limit fails to exist because of a jump, unbounded growth or oscillation.

How to use this limits intro test

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All AP Calculus AB chapters: AP Calculus AB math.

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