Evaluate infinity limit into the polynomial , Mathematics

Assignment Help:

Example   Evaluate following limits.

863_limit80.png

Solution

Here our first thought is probably to just "plug" infinity into the polynomial & "evaluate" every term to finds out the value of the limit.  This is pretty simple to illustrate what each term will do in the limit and so this look likes an obvious step.

Hence, let's see what we obtain if we do that.  As x approaches infinity, then x to a power can just get larger and the coefficient on each of the term (the first and third) will jsut make the term even larger. Hence, if we look at what each of the term is doing in the limit we get the following,

2012_Limit81.png

Now, we've obtained a small, although easily fixed, problem to deal along with. Probably we are tempted to say that the answer is zero (since we have infinity minus infinity) or possibly -∞ (as we're subtracting two infinities off of one infinity).  Though, in both of the cases we'd be wrong

Infinities only don't always behave as real numbers do while it comes to arithmetic.  Without more work there is no way to know what ∞ -∞ will be and hence we really have to be careful along with this kind of problem. 

Hence, we require a way to get around this problem.  What we'll do here is factor out the largest power of x out of the whole polynomial as given,

1597_limit83.png

Now for each terms we have,

2365_limit84.png

The first limit is obviously infinity and for the second limit we'll use the fact above on the previous two terms. Hence by busing Fact 2 from the previous section we see value of the limit will be,

Fact 2

If  p ( x ) = an xn + an-1 xn -1 + ....... + a 1x + a0 is a polynomial of degree n (that means  an  ≠ 0 )  then,

959_limit85.png

What this fact is actually saying is that while we go to take a limit at infinity for a polynomial then all we have to really do is look at the term along with the largest power and asks what that term is doing in the limit as the polynomial will have the similar behavior.

Let's now move into some more complexes limits.


Related Discussions:- Evaluate infinity limit into the polynomial

#rounding off, I am the least two digit number which round off to 100?

I am the least two digit number which round off to 100?

Estimate what percent of decrease for population, The population of Hamden ...

The population of Hamden was 350,000 in 1990. By 2000, the population had decreased to 329,000. What percent of decrease is this? First, ?nd out the number of residents who lef

Surds and logarithms, what are these all about and could i have some exampl...

what are these all about and could i have some examples of them please

Logorithms, log base 5 (3-2x) + log base 5 (2+x) = 1

log base 5 (3-2x) + log base 5 (2+x) = 1

Multiples, The sum of the smallest and largest multiples of 8 up to 60 is?

The sum of the smallest and largest multiples of 8 up to 60 is?

Ellpsoid, different kind of ellipsoid

different kind of ellipsoid

Volume, a data set has a mean of 3, a median of 4, and a mode of 5.which nu...

a data set has a mean of 3, a median of 4, and a mode of 5.which number must be in the data set-3,4,5?

Decision trees and sub sequential decisions, Decision Trees And Sub Sequent...

Decision Trees And Sub Sequential Decisions A decision tree is a graphic diagram of different decision alternatives and the sequence of events like if they were branches of a t

Math, a business is owned by three people.the first owns 1/12 of the busine...

a business is owned by three people.the first owns 1/12 of the business and the second owns 1/6 of the business. what fractional part of the business is owned by the third person

Expected value of perfect information, Expected Value of Perfect Informatio...

Expected Value of Perfect Information In the above problems we have used the expected value criterion to evaluate the decisions under the conditions of risk. But, as long as un

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd