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Tuesday, 26 March 2013

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
displaymath236,
for any a > 0.
Answer: Set tex2html_wrap_inline240 . We have
displaymath242. ln(a)
Clearly, we have
displaymath244.
Hence,
displaymath246
which translates into
displaymath236.

Example: Show that
displaymath250.
Answer: We will make use of the integral while the Hôpital Rule would have done a cleaner job. We have
displaymath252
so
displaymath254.
For tex2html_wrap_inline256 , we have tex2html_wrap_inline258, which is equivalent to tex2html_wrap_inline260 . Hence,
displaymath262.
But,
displaymath264.
Therefore, putting the stuff together, we arrive at
displaymath266.
Since,
displaymath268,
as n goes to tex2html_wrap_inline272 and tex2html_wrap_inline274 , the Pinching Theorem gives
displaymath250.

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 tex2html_wrap_inline282 .
As a direct application of the above limit, we get the next one:
Example: Show that
displaymath284.
Answer: Set tex2html_wrap_inline286 . We have
displaymath288.
Clearly, we have (from above)
displaymath244.
Hence,
displaymath246,
which translates into
displaymath284.

The next limit is extremely important and I urge the reader to be aware of it all the time.
Example: Show that
displaymath296,
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
displaymath300.
Do not worry about the domain of tex2html_wrap_inline302, since for large n, the expression tex2html_wrap_inline306 will be a positive number (close to 1). Consider the function
displaymath308
and f(0) = 1. Using the definition of the derivative of tex2html_wrap_inline312 , we see that f(x) is continuous at 0, that is, tex2html_wrap_inline316 . Hence, for any sequence tex2html_wrap_inline318 which converges to 0, we have
displaymath320.
Now, set
displaymath322.
Clearly we have tex2html_wrap_inline324 . Therefore, we have
displaymath326.
But, we have
displaymath328,
which clearly implies
displaymath330.
Since
displaymath332,
we get
displaymath334.

The next example, is interesting because it deals with the new notion of series.
Example: Show that
displaymath336
Answer: There are many ways to handle this sequence. Let us use calculus techniques again. Consider the function
displaymath338.
We have
displaymath340
and
displaymath342,
for any tex2html_wrap_inline344 . Note that for any tex2html_wrap_inline346 , we have
displaymath348,
hence
displaymath350,
which gives
displaymath352.
Since
displaymath354,
we get
displaymath356.
In particular, we have
displaymath358.
Therefore, since tex2html_wrap_inline360 , we must have
displaymath362.





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