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Okay, so I found this question in a text,

For a certain value of 'c', the given limit is finite & non-zero, and equal to 'l'. Then find 'l' & 'c'.

$$ \lim_{x \to \infty} [ (x^5 + 7x^4 +2)^c - x ] $$

To solve this problem, I thought that for the given limit to be finite, c must be equal to $\frac{1}{5}$, because if it's anything else, than the answer will tend to negative or positive infinity (because we have $x^5$ in the polynomial, and it can't have a power larger than 1).

Now I understand that this isn't exactly the best of ways to solve this question, so I would like to know how you would approach this question, and what would you generally do in cases like this?

PS: The answers are $ c = \frac{1}{5} $ and $ l = \frac{7}{5} $.

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AnonMouse
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1 Answers1

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The easiest method is to use Taylor expansion:

$$(x^5+7x^4+2)^c=x^{5c}\left(1+\frac7x+\frac2{x^5}\right)^c\sim_\infty x^{5c}\left(1+\frac{7c}x\right)$$ so taking $5c=1$ we see that the desired limit is $l=7c$.

user296113
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  • Thanks a lot! I just have one doubt, after concluding that $ x^{5c}\left(1+\frac{7c}x\right) $ is what we end up with, can we sufficiently say that only for $ 5c = 1 $ the limit in finite? – AnonMouse Jul 12 '16 at 17:11
  • Yes. We sub-struct $x$ and we get $x^{5c}-x+7c x^{5c-1}$ so if $5c\ne1$ the limit isn't finite. – user296113 Jul 12 '16 at 17:14