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Chapter 1: Problem 8

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow-4}(2 x+3) $$

Short Answer

Expert verified
The limit of the function \(2x + 3\) as \(x\) approaches -4 is -5.

Step by step solution

01

Identify the function

The function given is \(2x + 3\). We are asked to find the limit as \(x\) tends towards -4.
02

Substitute the value of x

In this case, substitute -4 for \(x\) in the function \(2x + 3\). This will give us the value of the function as \(x\) tends towards -4.
03

Compute the value

By substituting -4 for \(x\) in the function \(2x + 3\), we get \(2*-4 + 3 = -8 + 3 = -5\).

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Key Concepts

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

Evaluating Limits
Limits are fundamental in understanding how functions behave near a particular point. When we talk about evaluating limits, we mean finding the value that a function approaches as the input (in this case, \(x\)) gets closer to a specific number, termed as the limit point. In mathematical notation, we express this as \(\lim_{x \rightarrow a} f(x)\). This expression seeks the value that \(f(x)\) approaches as \(x\) nears \(a\).
**Approaching a Point**
  • If the terms of the function become arbitrarily close to a specific value as \(x\) approaches \(a\), then the limit exists.
  • If the terms of the function approach two different values from the positive and negative sides, the limit at that point does not exist.
Evaluating limits correctly is essential to understanding the continuity and behavior of functions, especially around critical points where the behavior might change drastically or be undefined.
Substitution Method
The substitution method is one of the simplest strategies used to find the limits of functions. It involves directly replacing the variable with the value it approaches unless there's an issue like division by zero. By simply placing the approaching value into the function, we can swiftly determine the limit.
**When to Use Substitution**
  • When the function is well-behaved and there is no danger of undefined operations at the point of interest.
  • Functions that are polynomial, rational (as long as the denominator isn't zero), and certain trigonometric functions at certain points.
In our example, substituting \(x = -4\) into the function \(2x + 3\) gives us \(-5\). This straightforward method confirms that \(\lim_{x \rightarrow -4} (2x + 3) = -5\), illustrating that the function smoothly reaches this value as \(x\) tends to \(-4\).
Continuous Functions
A function is continuous at a point \(x = a\) if the following three criteria are met:
  • The function is defined at \(x = a\); meaning \(f(a)\) exists.
  • The limit of the function as \(x\) approaches \(a\) exists.
  • The limit of the function as \(x\) approaches \(a\) is equal to the actual function value \(f(a)\); \(\lim_{x \rightarrow a} f(x) = f(a)\).
Continuous functions are crucial because they behave reliably without sudden jumps or breaks.
For the function \(2x + 3\), it is a linear polynomial and is continuous everywhere. Since linear functions do not break, the limit as \(x\) approaches any number, including \(-4\), can typically be found using direct substitution. This continuity means that the function's graph is a straight line, maintaining a consistent path everywhere it's defined.

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Most popular questions from this chapter

Testing for Continuity In Exercises \(77-84\) , describe the interval(s) on which the function is continuous. $$ f(x)=\left\\{\begin{array}{ll}{\frac{x^{2}-1}{x-1},} & {x \neq 1} \\ {2,} & {x=1}\end{array}\right. $$

Removable and Nonremovable Discontinuities Describe the difference between a discontinuity that is removable and one that is nonremovable. In your explanation, give examples of the following descriptions. (a) A function with a nonremovable discontinuity at \(x=4\) (b) A function with a removable discontinuity at \(x=-4\) (c) A function that has both of the characteristics described in parts (a) and (b)

Dirichlet Function Show that the Dirichlet function $$ f(x)=\left\\{\begin{array}{l}{0, \text { if } x \text { is rational }} \\ {1, \text { if } x \text { is irrational }}\end{array}\right. $$ is not continuous at any real number.

Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=\frac{|x+7|}{x+7} $$

Writing In Exercises 85 and \(86,\) use a graphing utility to graph the function on the interval \([-4,4] .\) Does the graph of the function appear to be continuous on this interval? Is the function continuous on \([-4,4] ?\) Write a short paragraph about the importance of examining a function analytically as well as graphically. $$ f(x)=\frac{x^{3}-8}{x-2} $$
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