Thank you very much 作者: 風之男 時間: 2021-6-11 08:52 PM
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).