Sec. 1.1 – 1.3 Name______
AP Calculus Practice Quiz – Limits Date______
Directions and Huge Hint: Don’t forget that you have four ways to examine every mathematical problem. When symbols do not make sense, you should be graphing and generating a number table as a matter of habit to try to figure out what is going on. Talking it out with yourself also helps!
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1. Evaluate the limit numerically: by creating a table that lists several values of f(x) for values of x near 0.
Thus we conclude that =
2. We can prove analytically that the limit in problem 1 exists, but not using the method. What other method of proof could we use? Hint: READ THE BOOK.
3. Another look at : Notice from the number table in problem 1 that when x values are “small”, . This is a great example of
a) local linearity b) instantaneous rate of change c) average rate of change
d) a secant line e) none of these
4. Sketch the graph of the function and then use the graph to answer the questions in parts a, b, and c. If the limit does not exist, write DNE and explain why the limit does not exist.
a) Find
b) Find c) Find
5. Sketch the graph of the function and then use the graph to answer the questions in parts a, b, and c. If the limit does not exist, write DNE and explain why the limit does not exist.
a) Find
b) (review): what are the asymptotes of f(x)?
6. Sketch the graph of the function and then use the graph to answer the questions in parts a, b, and c. If the limit does not exist, write DNE and explain why the limit does not exist.
a) Find
b)
7. Evaluate:
a) = b) = c) =
Explain the meaning of each expression and your answer in a, b, and c using simple, complete English sentences.
a)
b)
c)
8. What were the four “break-through” problems of the 17th century that formed the basis of contemporary calculus?
i)
ii)
iii)
iv)
9. Use the definition of a limit to prove that exists.
10. Explain the basic underlying ideas or concepts of the proof. A good explanation might include both a graph as well as a verbal explanation.
11. Explain what the following means using an English sentence or sentences:
If and if , then .
12. Find the limit: , if the limit exists. If it does not exist, writhe DNE and explain why the limit does not exist.
13. Find the limit: . Use an analytical (i.e., algebraic) method to alter the rational function into an equivalent form.
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last update: 9-12-06 sw