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

Calculus level 2, the first question should be done using the website given...

the first question should be done using the website given (www.desmos.com/calculator )and another good example to explain using the graph ( https://www.desmos.com/calculator/ydimzr

Linear approximation method for interpolation, Linear Approxi...

Linear Approximation Method This is a rough and ready method of interpolation and is best used when the series moves in predicted interval

Integration techniques, Integration Techniques In this section we are ...

Integration Techniques In this section we are going to be looking at several integration techniques and methods. There are a fair number of integration techniques and some wil

Solution to an initial value problem, S olve the subsequent IVP. dv/dt =...

S olve the subsequent IVP. dv/dt = 9.8 - 0.196v;               v(0) = 48 Solution To determine the solution to an Initial Value Problem we should first determine the gen

Higher-order derivatives, Higher-Order Derivatives It can be se...

Higher-Order Derivatives It can be seen that the derivative of a function is also a function. Considering f'x as a function of x, we can take the derivative

Discrete-time signals as energy or power signals, Classify the following di...

Classify the following discrete-time signals as energy or power signals. If the signal is of energy type, find its energy. Otherwise, find the average power of the signal. X 1

Help, draw a right angle isosceles triangle with 9 triangles in it

draw a right angle isosceles triangle with 9 triangles in it

Find out the surface area of the solid, Find out the surface area of the so...

Find out the surface area of the solid acquired by rotating y = √ (9-x 2 ), - 2 x 2 about the x-axis. Solution The formula that we'll be using here is, S = ∫ 2Πyds

Determine the properties and query are definable in datalog, We now focus o...

We now focus on the use of Datalog for defining properties and queries m graphs. (a) Suppose that P is some property of graphs definable in Datalog. Show drat P is preserved und

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