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Chapter 2: Problem 12

Exercise \(1-26:\) Find the limit, if it exists. $$ \lim _{x \rightarrow 1} 5 $$

Short Answer

Expert verified
The limit is 5.

Step by step solution

01

Understand the Concept of Constant Functions

The function given is a constant function, which means that its value does not change regardless of the input value of \( x \). In this case, the function is simply \( f(x) = 5 \).
02

Consider the Definition of a Limit

The limit of a function as \( x \) approaches a specific value is the value that the function gets closer and closer to as \( x \) gets closer to that specific value. For a constant function, the limit as \( x \) approaches any value is simply the constant value itself.
03

Apply the Limit Rule for Constant Functions

According to the rule for limits of constant functions, \( \lim_{x \to c} k = k \) for any constant \( k \) and any real number \( c \). Here, \( k = 5 \) and \( c = 1 \). Therefore, \( \lim_{x \to 1} 5 = 5 \).

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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 in mathematics. It is a function that always gives the same output, no matter the input value of the variable. In the realm of calculus, a constant function is described by a straight horizontal line on a graph. This line represents the constant value that the function always equals.
  • For example, if we have a constant function like \( f(x) = 5 \), it means that no matter what value you plug into \( x \), the output will always be 5.
  • This constancy makes them incredibly easy to work with, especially when determining limits.
Understanding constant functions is crucial as they form the foundation that leads into more complex types of functions.
Limit of a Function
The concept of a limit is central in calculus and can be thought of as the expected value that a function approaches as the input gets closer to a particular point. This means if you have a function \( f(x) \) and you want to know what happens as \( x \) approaches a certain value, the limit will tell you the approximate value that \( f(x) \) converges to.
  • The idea of a limit helps understand behaviors of a function at points that might be undefined in simple algebraic sense.
  • For example, the limit \( \lim_{x \to 1} f(x) \) tells us how \( f(x) \) behaves when \( x \) is super close to 1.
The use of limits allows us to study continuity, compute derivatives, and analyze integrals.
Limit Rule for Constant Functions
The limit rule for constant functions is incredibly straightforward. It states that the limit of a constant is simply the constant itself, regardless of the value that the variable is approaching.
In formal terms, \[ \lim_{x \to c} k = k \] where \( k \) is a constant and \( c \) is any real number. This rule emphasizes the simplicity of constant functions because no matter how \( x \) changes, the output remains unchanged.
  • In our exercise, the function given was \( f(x) = 5 \), which is a constant function.
  • So, when finding the limit as \( x \to 1 \), the result is still 5. There's no change needed; the constant preserves its value.
This rule simplifies calculating limits for constant functions and focuses our attention on more complex functions where variable behavior changes the output.

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Most popular questions from this chapter

Use the sandwich theorem to verify the limit. $$ \begin{aligned} &\lim _{x \rightarrow 0} \frac{|x|}{\sqrt{x^{4}+4 x^{2}+7}}=0\\\ &\text { (Hint: Use } f(x)=0 \text { and } g(x)=|x| \text { .) } \end{aligned} $$

Use theorems on limits to find the limit, if it exists. $$ \lim _{x \rightarrow 4} \sqrt[3]{x^{2}-5 x-4} $$

Use theorems on limits to find the limit, if it exists. $$ \lim _{x \rightarrow 1} \frac{x^{2}+x-2}{x^{5}-1} $$

Let \(f\) be defined as follows: $$ f(x)=\left\\{\begin{array}{ll} 0 & \text { if } x \text { is rational } \\ 1 & \text { if } x \text { is irrational } \end{array}\right. $$ Prove that for every real number \(a, \lim _{x \rightarrow a} f(x)\) does not exist.

Exer. \(39-42:\) Find all numbers at which \(f\) is continuous. $$ f(x)=2 x^{4}-\sqrt[3]{x}+1 $$
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