MATH103 April 2014
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Question 04 (c)

Evaluate $\lim _{n\to \infty }n\ln \left(1+{\frac {2}{n}}\right)$ if it exists, or show that it does not exist, otherwise.
Work must be shown for full marks. Simplify fully.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1

Use l’Hôpital’s rule.

Hint 2

Any product can be rewritten as a ratio.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
 If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
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Solution

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First, write the product as a ratio:
$n\ln \left(1+{\frac {2}{n}}\right)={\frac {\ln \left(1+{\frac {2}{n}}\right)}{\frac {1}{n}}}.$
Note that both the numerator and denominator in this ratio tend to $0$ as $n\to \infty$ (since $\ln(1+2/n)\to \ln(1)=0$). That is,
${\frac {\lim _{n\to \infty }\ln \left(1+{\frac {2}{n}}\right)}{\lim _{n\to \infty }{\frac {1}{n}}}}$
is an indeterminate form. Thus, we can apply l’Hôpital’s rule to get
$\lim _{n\to \infty }n\ln \left(1+{\frac {2}{n}}\right)=\lim _{n\to \infty }{\frac {\ln \left(1+{\frac {2}{n}}\right)}{\frac {1}{n}}}=\lim _{n\to \infty }{\frac {{\frac {d}{dn}}\ln \left(1+{\frac {2}{n}}\right)}{{\frac {d}{dn}}{\frac {1}{n}}}}.$
Now
${\frac {d}{dn}}\ln \left(1+{\frac {2}{n}}\right)={\frac {2n^{2}}{1+{\frac {2}{n}}}},$
while
${\frac {d}{dn}}{\frac {1}{n}}=n^{2}.$
Thus,
${\frac {{\frac {d}{dn}}\ln \left(1+{\frac {2}{n}}\right)}{{\frac {d}{dn}}{\frac {1}{n}}}}={\frac {\left({\frac {2n^{2}}{1+{\frac {2}{n}}}}\right)}{n^{2}}}={\frac {2}{1+{\frac {2}{n}}}}.$
Therefore,
$\lim _{n\to \infty }{\frac {{\frac {d}{dn}}\ln \left(1+{\frac {2}{n}}\right)}{{\frac {d}{dn}}{\frac {1}{n}}}}=\lim _{n\to \infty }{\frac {2}{1+{\frac {2}{n}}}}={\frac {2}{1+\lim _{n\to \infty }{\frac {2}{n}}}}={\frac {2}{1}}=2.$
Note: l’Hôpital’s rule is only applicable when its conditions are met by the numerator, the denominator, and even the ratio itself.

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