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Chapter 1: Problem 26

Find the limit. \(\lim _{x \rightarrow 0}(3 x-2)\)

Short Answer

Expert verified
The limit \(\lim _{x \rightarrow 0}(3 x-2)\) equals -2

Step by step solution

01

Understand the Problem

We are given the limit \(\lim _{x \rightarrow 0}(3 x-2)\) and asked to determine its value. This is a straightforward application of the properties of limits for linear functions.
02

Substitute the Limit Point

We replace \(x\) in the expression with the limit point, which is 0. So, the limit turns into: \((3 \cdot 0 - 2)\)
03

Evaluate the Expression

Evaluate the expression by performing the multiplication and subtraction: \(3 \cdot 0 - 2 = -2\)

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Key Concepts

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

Linear Functions
Linear functions are the simplest type of functions in algebra and calculus. They are generally expressed in the form of \( f(x) = ax + b \), where \(a\) and \(b\) are constants. These functions create straight-line graphs, representing a constant rate of change.
This means that as \(x\) increases or decreases, the function changes consistently due to the fixed slope \(a\).
  • Example: \(3x - 2\) is a linear function where the slope \(a\) is 3 and the y-intercept \(b\) is -2.
  • The graph of \(3x - 2\) will be a straight line crossing the y-axis at -2.
Understanding linear functions forms a basic building block for higher-level calculus concepts like limits. With linear functions, evaluating limits becomes straightforward because their behavior is predictable.
Limit Evaluation
Limit evaluation is a fundamental concept in calculus that examines the behavior of functions as they approach a specific point. Limits help us understand what value a function approaches as the input (\(x\)) approaches a certain value.
To evaluate a limit, substitute the approaching value into the function, if possible. This is known as direct substitution.
  • Direct substitution: Evaluate limits by directly replacing \(x\) with the approaching value in the function.
For instance, in the exercise \( \lim_{x \rightarrow 0} (3x - 2) \), substituting \(x = 0\) yields \(3 \times 0 - 2 = -2\).
This direct method often works for linear functions since they have no abrupt changes or discontinuities.
Properties of Limits
The properties of limits simplify the process of limit evaluation, especially for linear functions and other simple expressions.
Key properties include:
  • Limit of a constant: The limit of a constant is the constant itself.
  • Limit of a sum: The limit of a sum is the sum of the limits.
  • Limit of a product: The limit of a product is the product of the limits.
  • Substitution property: If a function is continuous at a point, the limit can be evaluated by substituting the point's value into the function.
Using these properties, calculating the limit \( \lim_{x \rightarrow 0} (3x - 2) \) becomes straightforward. Apply the properties individually by substituting \(x = 0\) to break down and solve the expression:
  • Apply the limit to each part: \( \lim_{x \rightarrow 0} 3x = 3 \times 0 \) and \( \lim_{x \rightarrow 0} -2 = -2 \).
  • Combine the results: resulting in \(0 - 2 = -2\).
Through these properties, we can efficiently find the limit without complex computations.

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Most popular questions from this chapter

Environmental cost The cost \(C\) (in millions of dollars) of removing \(x\) percent of the pollutants emitted from the smokestack of a factory can be modeled by $$C=\frac{2 x}{100-x}$$ (a) What is the implied domain of \(C ?\) Explain your reasoning. (b) Use a graphing utility to graph the cost function. Is the function continuous on its domain? Explain your reasoning. (c) Find the cost of removing \(75 \%\) of the pollutants from the smokestack.

Find the limit (if it exists). \(\lim _{\Delta x \rightarrow 0} \frac{\sqrt{x+2+\Delta x}-\sqrt{x+2}}{\Delta x}\)

The amounts (in billions of dollars) spent on prescription drugs in the United States from 1991 through 2005 (see figure) can be approximated by the model $$ d(t)=\left\\{\begin{array}{ll}{y=0.68 t^{2}-0.3 t+45,} & {1 \leq t \leq 8} \\\ {y=16.7 t-45,} & {9 \leq t \leq 15}\end{array}\right. $$ where \(t\) represents the year, with \(t=1\) corresponding to 1991. (a) Use a graphing utility to graph the function. (b) Find the amounts spent on prescription drugs in 1997 , \(2000,\) and 2004 .

Find the limit (if it exists). \(\lim _{x \rightarrow-1} \frac{x^{3}-1}{x+1}\)

Use a graphing utility to estimate the limit (if it exists). \(\lim _{x \rightarrow 2} \frac{x^{2}-5 x+6}{x^{2}-4 x+4}\)
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