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.

[Trigonometry] [Calculus]
[Geometry] [Algebra] [Differential Equations]
[Calculus] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

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

Copyright 1999-2017 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour