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Let p(x)be a nonzero polynomial function. Evaluatelimxp(x+1)p(x).

Short Answer

Expert verified

If p(x)=k=0akxkthen the value oflimxp(x+1)p(x)is1.

Step by step solution

01

Step 1. Given information. 

The given expression that needs to Evaluate islimxp(x+1)p(x).

02

Step 2. polynomial function.

Consider a polynomial,

p(x)=a0x0+a1x1+a2x2++anxnp(x)=k=0akxk

So the value of p(x+1)will be the following.

p(x+1)=k=0akx+1k

03

Step 3. Value of limx→∞p(x+1)p(x).

Determine the value of limxp(x+1)p(x).

limxp(x+1)p(x)=limxakx+1kakxk=limxx+1xk=limx1+1xk=1k+0=1

So the value oflimxp(x+1)p(x)is1.

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Most popular questions from this chapter

True/False:

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If ak0, then k=1akconverges.

(b) True or False: If k=1akconverges, then ak0.

(c) True or False: The improper integral 1f(x)dxconverges if and only if the series k=1f(k)converges.

(d) True or False: The harmonic series converges.

(e) True or False: If p>1, the series k=1k-pconverges.

(f) True or False: If f(x)0as x, then k=1f(k) converges.

(g) True or False: If k=1f(k)converges, then f(x)0as x.

(h) True or False: If k=1ak=Land {Sn}is the sequence of partial sums for the series, then the sequence of remainders {L-Sn}converges to 0.

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

k=0ekk=0ek

Use either the divergence test or the integral test to determine whether the series in Given Exercises converge or diverge. Explain why the series meets the hypotheses of the test you select.

k=112k3/4

Leila finds that there are more factors affecting the number of salmon that return to Redfish Lake than the dams: There are good years and bad years. These happen at random, but they are more or less cyclical, so she models the number of fish qkreturning each year as qk+1=(0.14(1)k+0.36)(qk+h), where h is the number of fish whose spawn she releases from the hatchery annually.

(a) Show that the sustained number of fish returning in even-numbered years approach approximately qe=3hk=10.11k.

(Hint: Make a new recurrence by using two steps of the one given.)

(b) Show that the sustained number of fish returning in odd-numbered years approaches approximately qo=6111hk=10.11k.

(c) How should Leila choose h, the number of hatchery fish to breed in order to hold the minimum number of fish returning in each run near some constant P?

If limkakbkWhereLis a positive finite number, what may we conclude about the two series?

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