I really don't know how to continue.. can u show me some more steps, please?View attachment 15967
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\(\displaystyle \left [\frac{\sqrt[n]{3} + \sqrt[n]{4}}{2} \right ] ^n\)
= \(\displaystyle (4)*\left [\frac{\sqrt[n]{\frac{3}{4}} + 1}{2} \right ] ^n\)
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Before I show more steps, please tell me:I really don't know how to continue.. can u show me some more steps, please?
I have studied both but I don't know how to use it in this example.Before I show more steps, please tell me:
What math class are you taking now (or have taken recently) - e.g.- precalculus, advanced algebra, calculus-I, etc.?
Have you studied Taylor's expansion formula?
Have you studied formula for Binomial expansion?
Please expand using Binomial expansion theorem (first few terms):I have studied both but I don't know how to use it in this example.
Again assumed things that I should not have assumed. Somehow I thought L'Hospital was not allowed (we had spate of those sadistic problems in near-past!)I would state:
[MATH]L=\lim_{n\to\infty}\left(\left(\frac{(3)^{\frac{1}{n}}+(4)^{\frac{1}{n}}}{2}\right)^n\right)[/MATH]
Taking the natural log of both sides, we eventually get:
[MATH]\ln(L)=\lim_{n\to\infty}\left(\frac{\ln\left(\dfrac{(3)^{\frac{1}{n}}+(4)^{\frac{1}{n}}}{2}\right)}{\dfrac{1}{n}}\right)[/MATH]
Now you have the indeterminate for 0/0 and may apply L'Hôpital's Rule.
(1 + x)n=Please expand using Binomial expansion theorem (first few terms):
(1 + x)n = ?
I would state:
[MATH]L=\lim_{n\to\infty}\left(\left(\frac{(3)^{\frac{1}{n}}+(4)^{\frac{1}{n}}}{2}\right)^n\right)[/MATH]
Taking the natural log of both sides, we eventually get:
[MATH]\ln(L)=\lim_{n\to\infty}\left(\frac{\ln\left(\dfrac{(3)^{\frac{1}{n}}+(4)^{\frac{1}{n}}}{2}\right)}{\dfrac{1}{n}}\right)[/MATH]
Now you have the indeterminate for 0/0 and may apply L'Hôpital's Rule.
First you should observe that:
[MATH]\ln(L)=\lim_{n\to\infty}\left(\frac{\ln\left(\dfrac{(3)^{\frac{1}{n}}+(4)^{\frac{1}{n}}}{2}\right)}{\dfrac{1}{n}}\right)[/MATH]
Now you have the indeterminate for 0/0 and may apply L'Hôpital's Rule.
Mark took you out of your misery - but please work through the intermediate steps.
It will help you to build "your character" (according to my UG calculus professor).