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Calculus/Limit law proof problem

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Expert: - 8/30/2009


Question

attached_pic
Hello! =)

I have a problem with the limit laws in Calculus. I don't understand one point in the product law proof...:S

I have attached a picture which clarifies the problem =) I drew a red square around the problem point. Where does this term come from and why this term... etc etc =)

thank you very much for reply in advance =)
Jonne


 

Answer
Questioner: Jonne
Country: Finland
Category: Calculus
Private: No
Subject: Limit law proof problem
Question:

attached_picHello! =)

I have a problem with the limit laws in Calculus. I don't understand one point in the product law proof...:S

I have attached a picture which clarifies the problem =) I drew a red square around the problem point. Where does this term come from and why this term... etc etc =)

thank you very much for reply in advance =)
Jonne
.................................
Hi, Jonne,

[Comment: I cannot make delta and epsilon; I will use d's and e's.]

You have not sent me the next page, which has the completion of the proof, but I suspect it contains something like this;

Now take  d = the smaller of  d1 and d2.

Remember,

d1 was the one that makes
                 e
| g(x) - M | < ----------
              2(1 + |L|)

and d2 makes
| g(x) - M | < 1

Then it will show that | fg - LM | < e.

The expression:
   e
----------
2(1 + |L|)

is not derived from something you saw earlier, it comes from something on the next page;  the author knows he will need it later, so he says:

"I can make | g(x) - M | AS SMALL AS I LIKE.  What do I like?  I like it this small:"
   e
----------
2(1 + |L|)

"Why do I like it?  Because it will come in handy later."

How did he know to write this and not:
   e
------------
73(1 + |L|)

or:

   e
------------
17(1 + |L|)

BECAUSE HE HAS DONE THIS BEFORE!

Why do you NOT like it?  I completely understand.

    Questioner's Rating
    Rating(1-10)Knowledgeability = 10Clarity of Response = 10Politeness = 10
    CommentThank you very much for your reply about the limit law =D I think I understood the idea now =D and sorry for not sending the second page, but your guess was correct about what it included in it =D best regards Jonne


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Paul Klarreich

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All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

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I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

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