   Chapter 10.3, Problem 111E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

(Compare Exercise 73 in Section 10.1.) Which indeterminate form results from lim x → + ∞ p ( x ) e x  if  p ( x ) is a polynomial? Numerically or graphically estimate these limits for various polynomials p ( x ) . What does this suggest about limits that result in p ( ∞ ) e ∞ ?

To determine

The indeterminate form of the limit limx+p(x)ex, if p(x) is a polynomial and also suggest about the limit that results in p()e.

Explanation

Given Information:

The limit is limx+p(x)ex.

Where p(x) is a polynomial

Consider the limit, limx+p(x)ex

Here, p(x) is a polynomial and a polynomial is always in the form of xn. So, when x approaches to , the polynomial always approach ±.

Now,

e>1

So, e=

Hence, the limit limx+p(x)ex is in the indeterminate form of ±.

For example:

Consider the polynomial p(x)=x2+2x+1

So, the limit can be written as:

limx+p(x)ex=limx+x2+2x+1ex

Now, apply continuity of closed-form theorem limxaf(x)=f(a):

limx+x2+2x+1ex=()2+2()+1e()=()2++1=

Here, is the indeterminate form.

Apply L

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