Differentiation and the limit notation

Question

I currently started studying some aspects of calculus for the first time in a very long time, and now I am struggling to understand some basic and silly concepts of calculus including the notations.

I have an example such as "Find $\lim_{n \to \infty} ( \log n^2 / n)$" -- which turns out to be $0$.

Now, is this question asking me to differentiate or I misunderstood it? If so, why is it using the limit notation rather than the differentiation notation: $d/dn$ or $f'(n)$. Also, are we always supposed to differentiate when we see something like "$\lim_{n \to \infty}$"?

Although this question might sound really silly, I would really appreciate some explanation.

Thanks in advance. :)

Note: I forgot to say that-- it seems to me that the book did differentiate the numerator and the denominator in the last step which is why I got confused.

lim n→∞(log n^2/n) =lim n→∞(2log n/n) = lim n→∞((2/ln 2) (ln n /n) ) = (2/ln 2) lim n→∞(1/n) = 0.

Answer 1

The question has nothing to do with differentiation; it is asking for the limit as $n$ goes to infinity of $\frac{\log n^2} n$.

If you are not comfortable with limits, as it appears, I suggest you read about limits before reading about derivatives. With the new information in the question, your book is suggesting you apply L'Hospital's Rule. That's a bad idea if you are struggling with the basic concepts.

Answer 2

Consider infinity as a concept not a number. A diverging geometric progression -it gets bigger and bigger and never stops. Now, if you are to investigate an expression with a parameter with 'value' of infinity, you know to evaluate the expression as the parameter gets bigger and bigger because you cannot set a parameter equal to a never-ending progression.

Consider how to determine the rate of change of something. You differentiate with respect to some spatial or temporal component. If you want to determine how fast something is going, you differentiate the equation of displacement with respect to time. Here you want to see how fast the numerator grows, compared to the denominator. In other words, you want to compare the rates of change.

$$\lim_{n \to \infty} \frac{\log (n^2)}{n}$$

$$ \frac{d}{dn} \frac{\log (n^2)}{n} = \frac{d}{dn}\left(\frac{2 log(n)}{n}\right) $$

Proceed to calculate using the quotient rule of differentiation:

$$\frac{n\frac{1}{n}-\log(n).1}{n^2} =\frac{1-log(n)}{n^2}$$

Nevertheless, logarithmic functions compress the value of a number, so log(n) will be less than n. This means that as n gets to be larger and larger, the denominator will make the fraction 'tend' towards zero. It may do so from the negative or positive 'side' of zero.

Check out l'hopital's rule for some interesting and informative mathematics.

You do not necessarily have to differentiate every time you see $\lim{n \to \infty}$, unless you are specifically interested in rate of change.