Use Higher Degree Approx for e^x about x = 0 to explain limit value

[yli]

Hi, I was wondering if someone could help me sort out my thoughts and perhaps, if possible start me off for this question:

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Use higher-degree approximations for \(\displaystyle e^x\) about \(\displaystyle x\, =\, 0\) to explain why this limit is "equal to infinity" for n = 1, 2, 3, .... Justify your answer, being careful to state any assumptions you are making.

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I know that e^x about x=0 is equal to one, but I am a bit confused about how to prove the limit is infinity in this way. Do I need to write out the polynomial and then insert infinity? Any help/clarification would be appreciated.

 

[barrick]

Hi, I was wondering if someone could help me sort out my thoughts and perhaps, if possible start me off for this question:

---

Use higher-degree approximations for \(\displaystyle e^x\) about \(\displaystyle x\, =\, 0\) to explain why this limit is "equal to infinity" for n = 1, 2, 3, .... Justify your answer, being careful to state any assumptions you are making.

---

I know that e^x about x=0 is equal to one, but I am a bit confused about how to prove the limit is infinity in this way. Do I need to write out the polynomial and then insert infinity? Any help/clarification would be appreciated.

Hi yli,

Could you tell us what limit you are talking about ? (That is not included in you attachment).

 

[tkhunny]

Normally, one would pick a large number, say "M", and prove that your expression ALWAYS exceeds this large number.

 

[yli]

Sorry, I thought it was already included, the limit is \[\frac{e^x}{x^n}\] when x approaches infinity and when n=1,2,3...

 

[tkhunny]

Oh, well, in that case, normally, one would pick a large number, say "M", and prove that your expression ALWAYS exceeds this large number. Let's see what you get.

 

[Dr.Peterson]

Sorry, I thought it was already included, the limit is \[\frac{e^x}{x^n}\] when x approaches infinity and when n=1,2,3...

You were told to "Use higher degree approximations for e^x about x=0 to explain why this limit is equal to infinity for n = 1, 2, 3, ... ."

Tell us what you have been taught about "higher degree approximations for e^x about x=0".

Your initial comment, "I know that e^x about x=0 is equal to one", suggests that you are not quite clear what this means. It is not about the value of e^x at x=0, but about a polynomial approximation centered at x=0.

Exactly what you can say to justify the conclusion will depend on exactly what you have learned. I notice that it says only, "Justify your answer, being careful to state any assumptions you are making," so it seems likely that you have not been taught quite enough to make a complete proof. Show us what you think.