Chapter 7: Q. 3 (page 639)

Let $p (x)$ be a nonzero polynomial function. Evaluate $\lim_{x \to \infty} \frac{p (x + 1)}{p (x)} .$

Short Answer

Expert verified

If $p (x) = \sum_{k = 0}^{\infty} a_{k} x^{k}$ then the value of $\lim_{x \to \infty} \frac{p (x + 1)}{p (x)} is 1 .$

Step by step solution

Step 1. Given information.

The given expression that needs to Evaluate is $\lim_{x \to \infty} \frac{p (x + 1)}{p (x)} .$

Step 2. polynomial function.

Consider a polynomial,

$p (x) = a_{0} x^{0} + a_{1} x^{1} + a_{2} x^{2} + \dots + a_{n} x^{n} p (x) = \sum_{k = 0}^{\infty} a_{k} x^{k}$

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

$p (x + 1) = \sum_{k = 0}^{\infty} a_{k} {(x + 1)}^{k}$

Step 3. Value of limx→∞p(x+1)p(x).

Determine the value of $\lim_{x \to \infty} \frac{p (x + 1)}{p (x)} .$

$\lim_{x \to \infty} \frac{p (x + 1)}{p (x)} = \lim_{x \to \infty} \frac{a_{k} {(x + 1)}^{k}}{a_{k} x^{k}} = \lim_{x \to \infty} {(\frac{x + 1}{x})}^{k} = \lim_{x \to \infty} {(1 + \frac{1}{x})}^{k} = 1^{k} + 0 = 1$

So the value of $\lim_{x \to \infty} \frac{p (x + 1)}{p (x)} is 1 .$

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Let $p (x)$ be a nonzero polynomial function. Evaluate $\lim_{x \to \infty} \frac{p (x + 1)}{p (x)} .$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. polynomial function.

Step 3. Value of limx→∞p(x+1)p(x).

One App. One Place for Learning.

Most popular questions from this chapter

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Let p(x)be a nonzero polynomial function. Evaluatelimx→∞p(x+1)p(x).

Short Answer

Step by step solution

Step 1. Given information.

Step 2. polynomial function.

Step 3. Value of limx→∞p(x+1)p(x).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Logic and Functions

Discrete Mathematics

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Let $p (x)$ be a nonzero polynomial function. Evaluate $\lim_{x \to \infty} \frac{p (x + 1)}{p (x)} .$