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[M2] limit Questipns

limit Questipns

Please help to answer the following questions

Thank you very much
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I doubt DSE actually asks a straight up limits question these days. Often it's asked in diff by first principles, but those are relatively easy.

When dealing with problems about limits, and you find an expression in indeterminate form (0/0, infty/infty), a trick that isn't taught is using L'Hopital's Rule. The idea is that you can differentiate the numerator and the denominator separately (if their limits exist), and the limit is still the same. Now these 2 problems don't look like 0/0 or infty/infty, but you can make them so. Search L'Hopital's Rule to see more info.
EDIT: WARNING! Do NOT use L'Hopital's Rule for diff. by first principles problems! It's a circular reasoning fallacy.

But suppose you don't know L'Hopital's Rule, how would you go about it? Here are some inspiration:

47 is a bit simpler. In fact you need to know this for diff. by first principles. When you see "sqrt + sqrt", the usual trick is to multiply and divide its conjugate.

Two inspirations for 55: (i) the stuff inside the parentheses can be factorized; and (ii) we only know one limit that's related to an exponential, and that's the number e (or e^x).

Answers in a bit.

EDIT:
Solution - https://www.docdroid.net/CdkD0K5/limits-pdf
I tried to not skip any steps, and added some remarks. If you still have queries, feel free to ask.

[ 本帖最後由 風之男 於 2021-6-11 10:14 PM 編輯 ]

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回覆 2# 風之男 的帖子

Thank you very much.
After your full explanation, we have got a very clear concept now.
What you have said is much better than our teacher's.

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