## Some Special Limits Here we will discuss some important limits that everyone should be aware of. They are very useful in many branches of science.

Example: Show using the Logarithmic function that ,

for any a > 0.

Answer: Set . We have . ln(a)

Clearly, we have .

Hence, which translates into .

Example: Show that .

Answer: We will make use of the integral while the Hôpital Rule would have done a cleaner job. We have so .

For , we have , which is equivalent to . Hence, .

But, .

Therefore, putting the stuff together, we arrive at .

Since, ,

as n goes to and , the Pinching Theorem gives .

The difficulty in this example was that both the numerator and denominator grow when n gets large. But, what this conclusion shows is that n grows more powerfully than .

As a direct application of the above limit, we get the next one:

Example: Show that .

Answer: Set . We have .

Clearly, we have (from above) .

Hence, ,

which translates into .

The next limit is extremely important and I urge the reader to be aware of it all the time.

Example: Show that ,

for any number a.

Answer: There are many ways to see this. We will choose one that involves a calculus technique. Let us note that it is equivalent to show that .

Do not worry about the domain of , since for large n, the expression will be a positive number (close to 1). Consider the function and f(0) = 1. Using the definition of the derivative of , we see that f(x) is continuous at 0, that is, . Hence, for any sequence which converges to 0, we have .

Now, set .

Clearly we have . Therefore, we have .

But, we have ,

which clearly implies .

Since ,

we get .

The next example, is interesting because it deals with the new notion of series.

Example: Show that Answer: There are many ways to handle this sequence. Let us use calculus techniques again. Consider the function .

We have and ,

for any . Note that for any , we have ,

hence ,

which gives .

Since ,

we get .

In particular, we have .

Therefore, since , we must have . [Trigonometry] [Calculus]
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Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard. Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996