Precalculus Introduction to Limits Test with Point Values (PDF)
This limits test for Precalculus covers introduction to limits (Common Core F-IF.B.6, F-IF.C.7d), with a free PDF to print.
Time and points
- Grade: Precalculus
- Chapter: Introduction to Limits
- Standards: F-IF.B.6, F-IF.C.7d
- Time: 45 minutes
- Total: 20 points
Limits test: questions
Question 1 – Average rate of change [2 pts]
Find the average rate of change of \(f(x)=x^{2}-3x\) from \(x=1\) to \(x=4\).
Question 2 – Limit from a table [2 pts]
The table shows \(f(x)=\frac{x^{2}-1}{x-1}\) near \(x=1\). Estimate \(\lim_{x\to1}f(x)\) and explain why \(f(1)\) is undefined.
Question 3 – Factor and cancel [2 pts]
Evaluate \(\lim_{x\to5}\frac{x^{2}-25}{x-5}\).
Question 4 – Rationalize [2 pts]
Evaluate \(\lim_{x\to9}\frac{\sqrt{x}-3}{x-9}\).
Question 5 – Read a graph [3 pts]
Use the graph. (a) Find \(\lim_{x\to1^{-}}f(x)\) and \(\lim_{x\to1^{+}}f(x)\). (b) Does \(\lim_{x\to1}f(x)\) exist? What is \(f(1)\)? (c) Is \(f\) continuous at \(1\)?
Question 6 – Limits at infinity [2 pts]
Evaluate (a) \(\lim_{x\to\infty}\frac{6x^{2}+x}{2x^{2}-3}\); (b) \(\lim_{x\to\infty}\frac{5x}{x^{2}+1}\).
Question 7 – Infinite limits [2 pts]
Find \(\lim_{x\to3^{+}}\frac{1}{x-3}\) and \(\lim_{x\to3^{-}}\frac{1}{x-3}\).
Question 8 – Step function [3 pts]
A delivery fee is \(\$5\) for \(0< w\le1\) lb, \(\$8\) for \(1< w\le2\) lb and \(\$11\) for \(2< w\le3\) lb. (a) Find \(\lim_{w\to2^{-}}C(w)\). (b) Find \(\lim_{w\to2^{+}}C(w)\). (c) Does \(\lim_{w\to2}C(w)\) exist? Explain.
Question 9 – Continuity [2 pts]
Find \(k\) so that \(f(x)=x+1\) for \(x< 2\) and \(f(x)=kx-3\) for \(x\ge2\) is continuous at \(2\).
Key facts behind this limits test
- Average rate of change: (f(b) – f(a))/(b – a), the secant slope.
- lim x to a of f(x) = L means f(x) approaches L as x approaches a from both sides.
- The limit can exist even if f(a) is undefined (a hole).
- Two-sided limit exists only if the left and right limits are equal.
- f is continuous at a if f(a) exists, the limit exists and they are equal.
How to use this limits 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 Precalculus chapters: Precalculus math.
