AP Calculus AB Evaluating Limits Algebraically Test with Point Values (PDF)

AP Calculus AB math test: Evaluating Limits Algebraically – Maths PDF

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This limit algebra test for AP Calculus AB covers evaluating limits algebraically (Common Core AP CALC AB 1.5, AP CALC AB 1.6, AP CALC AB 1.7, AP CALC AB 1.8), with a free PDF to print.

This test checks evaluating limits with substitution, factoring, conjugates, special trig limits and the Squeeze Theorem (AP Calculus AB, 1.5, 1.6, 1.7, 1.8). 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: Evaluating Limits Algebraically
Standards: AP CALC AB 1.5, AP CALC AB 1.6, AP CALC AB 1.7, AP CALC AB 1.8
Time: 45 minutes
Total: 20 points

Limit algebra test: questions

Question 1 – Substitution and limit laws [2 pts]

(a) Find \(\lim_{x\to -1} (x^{3}-4x+2)\). (1 point) (b) If \(\lim_{x\to 2} f(x)=6\) and \(\lim_{x\to 2} g(x)=3\), find \(\lim_{x\to 2}[f(x)/g(x)+2g(x)]\). (1 point)

Question 2 – Factoring [2 pts]

Find \(\lim_{x\to 5} \frac{x^{2}-3x-10}{x-5}\).

Question 3 – Rationalizing [3 pts]

The graph of (√(x + 25) − 5)/x.

Find \(\lim_{x\to 0} \frac{\sqrt{x+25}-5}{x}\).

Question 4 – Trigonometric limit [2 pts]

Find \(\lim_{x\to 0} \frac{\sin 6x}{2x}\).

Question 5 – Squeeze Theorem [3 pts]

A function between 2 − x² and 2 + x².

The graph shows a function \(h\) with \(2-x^{2}\le h(x)\le 2+x^{2}\) for all \(x\). Find \(\lim_{x\to 0} h(x)\) and name the theorem that justifies it. (3 points)

Question 6 – Difference quotient limit [2 pts]

For \(f(x)=2x^{2}+x\), find \(\lim_{h\to 0}\frac{f(1+h)-f(1)}{h}\).

Question 7 – Hole in a graph [3 pts]

The graph of g(x) = (x² + x − 6)/(x − 2).

Let \(g(x)=\frac{x^{2}+x-6}{x-2}\) with the graph shown. (a) Is \(g(2)\) defined? (1 point) (b) Find \(\lim_{x\to 2} g(x)\) algebraically. (1 point) (c) Explain how the graph shows your answer. (1 point)

Question 8 – Cost per item [3 pts]

A school store models the average cost per hoodie in dollars as \(C(n)=\frac{n^{2}-100}{n-10}\) for an order of \(n\) hoodies. (a) Find \(C(12)\). (1 point) (b) Find \(\lim_{n\to 10} C(n)\). (2 points)

Key facts behind this limit algebra test

Key facts

Direct substitution works when the function is defined and continuous at the target.
The form 0/0 means “simplify more”, not “the limit is 0” and not “does not exist”.
Multiply by the conjugate to remove a square root from a limit.
lim sin(x)/x = 1 and lim (1 – cos x)/x = 0 as x approaches 0 (radians).

How to use this limit algebra 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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