Other Indeterminate Quotient Forms

We have discussed the indeterminate quotient form tex2html_wrap_inline168 . But there some indeterminate quotient forms similar to this one. Indeed, since

displaymath170

then one can see that the indeterminate quotient forms

displaymath172

as well as the indeterminate forms

displaymath174

may be easily converted to the indeterminate quotient form tex2html_wrap_inline168 . So first notice that Hôpital's rule is still valid when dealing with the indeterminate quotient form tex2html_wrap_inline178 . Also worth to mention that the point a may be finite or infinite, Hôpital's rule still applies.

Example. Find the limit

displaymath182

Answer. Fix tex2html_wrap_inline184 . We have

displaymath186

Hence

displaymath188

We will use Hôpital's rule. We have

displaymath190

So clearly we will keep to use Hôpital's rule n-times to get to the function

displaymath192

Since

displaymath194

Therefore, we have

displaymath196

Note that this limit implies that, though both functions are very large when tex2html_wrap_inline198 , the exponential function tex2html_wrap_inline200 is more powerful than the power function tex2html_wrap_inline202 (in fact more powerful than any polynomial function). We will write

displaymath204

Example. Find the limit

displaymath206

Answer. Fix tex2html_wrap_inline208 . We have

displaymath210

Hence

displaymath212

We will use Hôpital's rule. We have

displaymath214

Hence we have

displaymath216

which implies

displaymath218

Clearly this example implies that

displaymath220

Putting the two examples together we conclude that

displaymath222

when tex2html_wrap_inline198 .

The indeterminate forms tex2html_wrap_inline226
The main idea behind these indeterminate forms is to transform 0 into tex2html_wrap_inline228 (the tex2html_wrap_inline230 depends on whether we have 0+ or 0-), or transform tex2html_wrap_inline236 into tex2html_wrap_inline238 . This will lead to the indeterminate quotient forms

displaymath240

Practically, you will be given a product f(x)g(x) where one function goes to 0 while the other one goes to tex2html_wrap_inline236 . So you will use the following algebraic manipulations

displaymath246

Example. Find the limit

displaymath248

Answer. We have

displaymath250

Hence

displaymath252

Rewrite the given expression into

displaymath254

Computing the limit we will find

displaymath256

Here we have a choice. We may use Hôpital's rule or Taylor Polynomials. In any case, Hôpital's rule is not bad to use in this case. Indeed, we have

displaymath258

Since

displaymath260

we conclude that

displaymath262

Note that when tex2html_wrap_inline264 , we have

displaymath266

and

displaymath268

which imply

displaymath270

May be this is easier, what do you think???

Example. Find the limit

displaymath272

Answer. We have again

displaymath274

We will ask you to check that whether you take

displaymath276

or

displaymath278

The calculations are not easy. Here let us show how some tricks may help. First switch from x into t = 1/x. We will have

displaymath284

Note that tex2html_wrap_inline286 when tex2html_wrap_inline288 . Next, we use

displaymath290

(when tex2html_wrap_inline292 ), to get

displaymath294

We already proved that

displaymath296

Therefore, we have

displaymath298

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

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

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