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Chapter 2: Problem 15

Evaluate the following limits. \(\lim _{x \rightarrow 6} 4\)

Short Answer

Expert verified
Answer: The value of the limit \(\lim_{x \rightarrow 6} 4\) is 4.

Step by step solution

01

Identify the constant function

The function we are evaluating the limit of is \(f(x) = 4\), which is a constant function.
02

Evaluate the limit

We have \(\lim_{x \rightarrow 6} 4 = 4\) because the value of a constant function doesn't change as the input approaches any particular value.
03

Provide the answer

The limit \(\lim_{x \rightarrow 6} 4\) is equal to 4.

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Key Concepts

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

Constant Function
A constant function is one of the simplest types of functions that you'll come across in mathematics. It is a function where the output or value remains the same for any input. For example, in the function given in the problem, \( f(x) = 4 \), no matter what value of \( x \) you substitute into the function, the result will always be 4.

To understand constant functions better, here are some key points to keep in mind:
  • The graph of a constant function is a horizontal line on the coordinate plane.
  • Since the function's value does not depend on \( x \), this means the function has no slope, and its rate of change is zero.
  • No matter what number you're approaching on the \( x \)-axis, the function's value will always remain the same.
Constant functions are frequently used as solutions to questions like the one you've seen, due to their straightforward nature. This makes solving problems involving limits with constant functions quite easy and predictable.
Evaluating Limits
Evaluating limits is a fundamental concept in calculus, which involves finding the value that a function approaches as the input approaches a certain value. In the provided exercise, we evaluated the limit of the constant function \( \lim_{x \rightarrow 6} 4 \).

Key points about evaluating limits:
  • If the function is constant, like \( f(x) = 4 \), the limit is simply that constant value, because the function does not change.
  • The limit tells us how the function behaves near a certain point, but not necessarily at that point. However, for constant functions, it behaves the same at all points.
  • Understanding how to evaluate limits is particularly crucial when dealing with more complex functions, where the behavior could change as \( x \) approaches different values.
The concept of limits helps set the groundwork for more advanced topics in calculus, such as derivatives and integrals. In this example, the problem is straightforward due to the simplicity of a constant function.
Approaching a Value
The phrase 'approaching a value' is a crucial part of understanding limits. It refers to the behavior of a function as the input gets closer and closer to a particular number. In this exercise, we focused on \( x \rightarrow 6 \).

Here's what you should know about approaching a value:
  • For constant functions, the output doesn't depend on the input, so it stays constant no matter what \( x \) approaches. That's why \( \lim_{x \rightarrow 6} 4 = 4 \).
  • In more complex functions, the closer \( x \) gets to the value, the more the function value tells us about its behavior, which can help in making better predictions or understandings.
  • Approaching a value is not the same as actually reaching that value, which can be especially relevant in situations involving points of discontinuity or undefined values.
Understanding the concept of approaching a value is key to mastering limits. It's the basis for exploring more advanced mathematical ideas and solving more challenging calculus problems.

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Most popular questions from this chapter

Calculate the following limits using the factorization formula \(x^{n}-a^{n}=(x-a)\left(x^{n-1}+x^{n-2} a+x^{n-3} a^{2}+\cdots+x a^{n-2}+a^{n-1}\right)\) where \(n\) is a positive integer and a is a real number. \(\lim _{x \rightarrow a} \frac{x^{5}-a^{5}}{x-a}\)

A monk set out from a monastery in the valley at dawn. He walked all day up a winding path, stopping for lunch and taking a nap along the way. At dusk, he arrived at a temple on the mountaintop. The next day the monk made the return walk to the valley, leaving the temple at dawn, walking the same path for the entire day, and arriving at the monastery in the evening. Must there be one point along the path that the monk occupied at the same time of day on both the ascent and descent? (Hint: The question can be answered without the Intermediate Value Theorem.) (Source: Arthur Koestler, The Act of Creation.)

Assume the functions \(f, g,\) and \(h\) satisfy the inequality \(f(x) \leq g(x) \leq h(x)\) for all values of \(x\) near \(a,\) except possibly at \(a .\) Prove that if \(\lim _{x \rightarrow a} f(x)=\lim _{x \rightarrow a} h(x)=L\) then \(\lim _{x \rightarrow a} g(x)=L\).

Let $$g(x)=\left\\{\begin{array}{ll}x^{2}+x & \text { if } x<1 \\\a & \text { if } x=1 \\\3 x+5 & \text { if } x>1. \end{array}\right.$$ a. Determine the value of \(a\) for which \(g\) is continuous from the left at 1. b. Determine the value of \(a\) for which \(g\) is continuous from the right at 1. c. Is there a value of \(a\) for which \(g\) is continuous at 1? Explain.

Let \(f(x)=\frac{2 e^{x}+10 e^{-x}}{e^{x}+e^{-x}} .\) Analyze \(\lim _{x \rightarrow 0} f(x), \lim _{x \rightarrow-\infty} f(x),\) and \(\lim _{x \rightarrow \infty} f(x) .\) Then give the horizontal and vertical asymptotes of \(f .\) Plot \(f\) to verify your results.
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