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Chapter 1: Problem 16

Finding a limit In Exercises \(5-22,\) find the limit. $$\lim _{x \rightarrow 0}(3 x-2)^{4}$$

Short Answer

Expert verified
The limit as x approaches 0 of \((3x-2)^4\) is 16.

Step by step solution

01

Substitution

The first step is to substitute 0 for x in the function \((3x-2)^4\) to check if the function is defined at x=0. After substitution we get \((3(0)-2)^4=(-2)^4\).
02

Simplify the Result

The second step is to simplify \((-2)^4\). This will yield 16.
03

Final Answer

Since the function \((3x-2)^4\) is defined and continuous at x=0, the limit as x approaches 0 is the function's value at x=0, which is 16.

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Key Concepts

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

Limit Substitution
Limit substitution is the initial step taken to find the limit of a function as a variable approaches a specific value. By directly substituting the target value of the variable into the function, we check if the function gives a well-defined result. In our exercise, we attempted to find the limit of \((3x-2)^4\) as \(x\) approaches 0.
  • We substitute 0 for \(x\) to get \((3*0-2)^4 = (-2)^4\).
  • If this substitution leads to a numerical value without any undefined operations, like division by zero, it indicates that the function behaves well around that input.
This method is straightforward and confirms that the function is defined at that particular point. If it's well-defined, the substituted value is often the limit.
Continuous Functions
A continuous function means its graph is unbroken; you can draw it without lifting your pencil. At any specific point, you can substitute a value into the function, and it returns a meaningful, non-infinite result. In our case, \((3x-2)^4\) is considered continuous at \(x = 0\).
  • Checking for continuity involves seeing that the function doesn't have any breaks or jumps at the point in question.
  • If a function is continuous at a specific point and defined at that point, its limit as \(x\) approaches that value equals the function value at the point.
Jacobians of limiting these conditions, like in our example, allow easy calculation because we can directly substitute and get a trustworthy limit figure. Continuity reassures us that the process won't spring any surprises.
Limit Simplification
Simplifying a limit expression is an essential step to finding a straightforward numerical value. After substitution, simplify the operations to solve the problem. From \((-2)^4\) to 16, the operation breaks down as follows:
  • We substitute 0 for \(x\) in \((3x-2)^4\) to make it \((-2)^4\).
  • Knowing that any negative base raised to an even power results in a positive value, we calculate \((-2)^4 = 16\).
Employing basic arithmetic laws here aids our understanding and provides a clear result. Simplification helps strip down the complex function to its core output, making it easier to interpret the limit.

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Most popular questions from this chapter

Estimating a Limit Consider the function \(f(x)=(1+x)^{1 / x}\) Estimate $$\lim _{x \rightarrow 0}(1+x)^{1 / x}$$ by evaluating \(f\) at \(x\) -values near \(0 .\) Sketch the graph of \(f\)

Testing for Continuity In Exercises \(75-82,\) describe the interval(s) on which the function is continuous. $$f(x)=\cos \frac{1}{x}$$

Think About It When using a graphing utility to generate a table to approximate $$\lim _{x \rightarrow 0} \frac{\sin x}{x}$$ a student concluded that the limit was 0.01745 rather than 1 . Determine the probable cause of the error

True or False? In Exercises \(105-110\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If $$\lim _{x \rightarrow c} f(x)=L$$ and \(f(c)=L,\) then \(f\) is continuous at \(c\)

$$\lim _{x \rightarrow 9^{-}} \frac{6}{9-x}=\infty$$
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