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Limits at Innity
In this section, Ill discuss proofs for limits of the form lim f (x). They are like  proofs, though the
x
setup and algebra are a little dierent.
Recall that lim f (x) = L means that for every > 0, there is a such that if
xc
>
Section 8: Working with the Denitions
8. Working with the Denitions
toc
We now come to that point in our discussion when we look at the
rigorous denition of limit. Once again, let me remind you of our only
description of limit: Pedestrian Description.
8.1
MULTIVARIABLE CALCULUS PRACTICE MIDTERM 1 SOLUTIONS
1.
(5 points ) If v w = 1, 1, 1 and v w = 2, nd the angle between v and w.
Solution: Let be the angle in question. By using the formula for the magnitude of cross
product (Theorem 3b on page 758), we obt
Section 3 Sequences and Limits, Continued.
Lemma 3.6 Let cfw_an nN be a convergent sequence for which an = 0 for all
n N and limit = 0. Then there exists N N such that

3 
an 
2
2
for all n N .
In particular this result ensures that if the limit is