/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 How is \(\lim _{x \rightarrow a}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 1

How is \(\lim _{x \rightarrow a} f(x)\) calculated if \(f\) is a polynomial function?

Short Answer

Expert verified
To find the limit of a polynomial function as x approaches a given value a, apply the properties of the limit such as the limit of a constant function c as x approaches any value a is c, the limit of x as x approaches a value a is a, and the limit of a product of functions is equal to the product of their limits. Then, compute the limit of individual terms, sum them up to get the overall limit of the polynomial function, and finally simplify the expression to find the limit of the given polynomial function as x approaches a value a.

Step by step solution

01

Definition of a Polynomial Function

A polynomial function is a function of the form $$ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 $$ where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants and \(n\) is a non-negative integer.
02

Properties of the Limit of a Polynomial Function

Limit of a polynomial function has the following properties: 1. The limit of a constant function c as x approaches any value a is c. 2. The limit of x as x approaches a value a is a. 3. The limit of a polynomial function as x approaches a, behaves as a sum of its individual terms. 4. The limit of a product of functions is equal to the product of their limits. Using these properties, we can find the limit of a polynomial function as x approaches a certain value a.
03

Identify the polynomial function and the value of a

Write down the given polynomial function \(f(x)\) and the value of x that it approaches (a).
04

Apply limit properties

For the given polynomial function \(f(x)\), apply the limit properties mentioned above, to get the limit of each individual term as x approaches a.
05

Compute the limit of individual terms

For each term, compute the limit as x approaches a using the limit properties.
06

Sum the limits of individual terms

Once you have calculated the limits of each individual term, sum them up to get the overall limit of the polynomial function: $$ \lim _{x \rightarrow a} f(x) = \lim _{x \rightarrow a} (a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0) $$
07

Simplify and conclude

Finally, simplify the overall limit expression to find the limit of the given polynomial function as x approaches a value a.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

We say that \(\lim _{x \rightarrow \infty} f(x)=\infty\) if for any positive number \(M,\) there is \(a\) corresponding \(N>0\) such that $$f(x)>M \quad \text { whenever } \quad x>N$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{x}{100}=\infty$$

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$g(x)=e^{1 / x}$$

Evaluate the following limits. $$\lim _{x \rightarrow 1^{-}} \frac{x}{\ln x}$$

A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence \(\\{2,4,6,8, \ldots\\}\) is specified by the function \(f(n)=2 n\), where \(n=1,2,3, \ldots .\) The limit of such a sequence is \(\lim _{n \rightarrow \infty} f(n)\), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences, or state that the limit does not exist. \(\left\\{2, \frac{3}{4}, \frac{4}{9}, \frac{5}{16}, \ldots\right\\},\) which is defined by \(f(n)=\frac{n+1}{n^{2}},\) for \(n=1,2,3, \ldots\)

Use the following definition for the nonexistence of a limit. Assume \(f\) is defined for all values of \(x\) near a, except possibly at a. We say that \(\lim _{x \rightarrow a} f(x) \neq L\) if for some \(\varepsilon>0\) there is no value of \(\delta>0\) satisfying the condition $$|f(x)-L|<\varepsilon \quad \text { whenever } \quad 0<|x-a|<\delta$$ Let $$f(x)=\left\\{\begin{array}{ll} 0 & \text { if } x \text { is rational } \\ 1 & \text { if } x \text { is irrational. } \end{array}\right.$$ Prove that \(\lim _{x \rightarrow a} f(x)\) does not exist for any value of \(a\). (Hint: Assume \(\lim _{x \rightarrow a} f(x)=L\) for some values of \(a\) and \(L\) and let \(\varepsilon=\frac{1}{2}\).)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.