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Chapter 2: Problem 14

Evaluate the following limits. \(\lim _{x \rightarrow 2}(-3 x)\)

Short Answer

Expert verified
Answer: The limit as \(x\) approaches \(2\) of the function \(-3x\) is \(-6\).

Step by step solution

01

Identify the function and the limit

The function given is \(f(x) = -3x\), and we are asked to find the limit as \(x\) approaches \(2\). So, we need to find \(\lim_{x \rightarrow 2}(-3x)\).
02

Evaluate the function at the limit point

Since this is a linear function, the limit will equal the value of the function at the point where \(x = 2\). So, we need to find the value of \(f(2)\). \(f(2) = -3(2) = -6\)
03

State the limit

The limit as \(x\) approaches \(2\) of the function \(-3x\) is the same as the value of the function at the point where \(x = 2\). Thus, \(\lim_{x \rightarrow 2}(-3x) = -6\).

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Key Concepts

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

Evaluation of Limits
The evaluation of limits is a fundamental concept in calculus. It helps us determine the behavior of a function as the input approaches a particular value.When evaluating limits, the first step is to identify the function and the point at which we are finding the limit. For instance, in our exercise, the function is \(-3x\), and we want to find the limit as \(x\) approaches \(2\).

Methods to evaluate limits include:
  • Direct substitution: Substitute the value into the function if it is continuous at that point.
  • Factorization: Useful when direct substitution leads to an indeterminate form, like \(0/0\).
  • Rationalization: This involves multiplying by a conjugate to simplify the function and resolve indeterminate forms.
In this exercise, using direct substitution is appropriate since the function is linear and continuous. Thus, the limit as \(x\) approaches \(2\) is just the function value at \(x=2\), which is \(-6\).
Linear Functions
Linear functions are among the simplest and most important types of functions in calculus. They have the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants.The key characteristics of linear functions include:
  • Their graphs are straight lines.
  • The slope \(m\) determines the steepness and direction of the line.
  • The y-intercept \(b\) is where the line crosses the y-axis.
The function \(-3x\) in our exercise is a linear function with slope \(-3\) and y-intercept \(0\).

Linear functions are continuous everywhere, which means evaluating limits at any point is straightforward. Simply substitute the point into the function. This is why, in our exercise, to find \(\lim_{x \rightarrow 2}(-3x)\), we just substitute \(x = 2\) into \(-3x\) to get \(-6\).
Calculus Basics
Understanding the basics of calculus is essential for evaluating limits and solving a wide range of mathematical problems. Calculus is primarily focused on two concepts: differentiation and integration.Differentiation involves finding the rate at which something changes.
  • The derivative of a function provides the slope of the tangent line at any point on the function.
  • In our case with the linear function \(-3x\), the derivative is constant \(-3\), indicating a constant rate of change.
Integration, on the other hand, deals with the accumulation of quantities, such as areas under curves.
  • While not directly used in evaluating limits, understanding integration helps in understanding the entire calculus framework.
Calculus basics also include understanding limits, as they lay the foundation for both differentiation and integration. The concept of limits allows us to handle functions that are not always easy or possible to evaluate at a given point directly, providing a deeper insight into the behavior of functions at particular points.

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Most popular questions from this chapter

Use analytical methods to identify all the asymptotes of \(f(x)=\frac{\ln \left(9-x^{2}\right)}{2 e^{x}-e^{-x}} .\) Then confirm your results by locating the asymptotes with a graphing utility.

Let \(f(x)=\frac{|x|}{x} .\) Then \(f(-2)=-1\) and \(f(2)=1 .\) Therefore \(f(-2)<0

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Evaluate the following limits, where \(c\) and \(k\) are constants. \(\lim _{x \rightarrow 2}(5 x-6)^{3 / 2}\)
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