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Chapter 1: Problem 26

Algebraically determine the limits. $$ \lim _{x \rightarrow 0.5}\left(10 x^{2}+8 x+6\right) $$

Short Answer

Expert verified
The limit is 12.5.

Step by step solution

01

Understand The Problem

We are given a limit problem where we need to find the limit of a polynomial function as \( x \) approaches a specific value, 0.5. Our function is \( f(x) = 10x^2 + 8x + 6 \).
02

Identify The Function's Limit

Since the function \( f(x) \) is a polynomial, it is continuous everywhere. Therefore, the limit of the polynomial as \( x \) approaches any point \( c \) is simply the value of the polynomial evaluated at \( x = c \).
03

Substitute And Solve

To find the limit as \( x \rightarrow 0.5 \), substitute \( x = 0.5 \) into the function:\[\lim_{x \to 0.5} (10x^2 + 8x + 6) = 10(0.5)^2 + 8(0.5) + 6\]Calculate each term:1. \( 10(0.5)^2 = 10 \times 0.25 = 2.5 \)2. \( 8(0.5) = 4 \)3. \( 6 = 6 \)Add these values together:\[2.5 + 4 + 6 = 12.5\]
04

Conclude The Solution

Since the function \( f(x) = 10x^2 + 8x + 6 \) is continuous and we have evaluated it at \( x = 0.5 \), we conclude that the limit of \( f(x) \) as \( x \rightarrow 0.5 \) is 12.5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions
Polynomial functions are expressions involving variables raised to whole-number exponents, typically written in the form \[a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\], where each \(a\) represents coefficients and \(n\) is a non-negative integer. Here are some key characteristics of polynomial functions:
  • The degree of the polynomial is the highest power of the variable in the expression.
  • Each term in the polynomial is a monomial, such as \(10x^2\), \(8x\), or \(6\) in our example.
  • Polynomial functions are smooth and continuous, meaning there are no breaks, holes, or jumps in their graphs.
Due to their smooth nature, polynomial functions are widely used in calculus, making them easier to work with, especially when we talk about limits, derivatives, and integrals.
Limits
Limits help us understand the behavior of functions as they approach a specific point. In calculus, the limit is a fundamental concept that describes how a function behaves as the input approaches a given value.For the exercise provided, we found the limit of the polynomial function \(f(x) = 10x^2 + 8x + 6\) as \(x\) approaches 0.5.To determine the limit of a polynomial function as \(x\) approaches a specific point:
  • Identify the value to which \(x\) is approaching, in this case, 0.5.
  • Substitute this value into the polynomial, replacing \(x\). "Evaluating at a point" is a straightforward process with polynomial functions due to their continuity.
  • Calculate the resulting expression to get the limit value.
In our case, substituting gives us 12.5. Limits are especially handy for functions not as straightforward as polynomials, helping identify behaviors near specific points.
Continuity
Continuity in a function means that the graph of the function is an unbroken line or curve. A continuous function has no jumps, breaks, or holes. This seamless nature is why polynomial functions are considered continuous everywhere.To check for continuity at a point:
  • The function must be defined at the point.
  • The limit of the function as it approaches the given point must exist.
  • The value of the function at the point must equal the limit as \(x\) approaches that point.
In our exercise, the function \(f(x) = 10x^2 + 8x + 6\) is continuous everywhere because all polynomial functions meet these criteria automatically.Understanding continuity helps in simplifying limit problems because, for continuous functions, the limit at \(x = c\) is simply \(f(c)\). This property makes it easier to compute limits and understand function behaviors at various points.

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Most popular questions from this chapter

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For each of the functions, state the amplitude, period, average value, and horizontal shift. \(\quad p(x)=235 \sin (300 x+100)-65\)

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