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Chapter 2: Problem 1

Write an equation that expresses the fact that a function \(f\) is continuous at the number 4 .

Short Answer

Expert verified
\( \lim_{{x \to 4}} f(x) = f(4) \)

Step by step solution

01

Understanding Continuity

To express that a function \( f \) is continuous at a specific point, we need to ensure that certain mathematical conditions are met. Specifically, a function \( f \) is said to be continuous at \( x = a \) if the limit of \( f(x) \) as \( x \) approaches \( a \) is equal to the function value at \( a \), i.e., \( f(a) \).
02

Setting the Specific Point

Given that we need to express continuity at the number 4, we substitute \( a = 4 \) into our definition of continuity. The condition for continuity then becomes that the limit of \( f(x) \) as \( x \) approaches 4 should be equal to \( f(4) \).
03

Writing the Continuity Equation

Formally, for a function \( f \) to be continuous at \( x = 4 \), we write the equation: \[ \lim_{{x \to 4}} f(x) = f(4) \] This equation captures the essence that as \( x \) gets arbitrarily close to 4, \( f(x) \) gets arbitrarily close to \( f(4) \), ensuring continuity at \( x = 4 \).

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Key Concepts

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

Limits
Understanding limits is crucial to grasping the idea of continuity. In mathematics, a limit describes the behavior of a function as the input approaches a particular point. When we say the limit of a function \( f(x) \) as \( x \) approaches some value \( a \) is \( L \), written as \( \lim_{{x \to a}} f(x) = L \), it means that the values of \( f(x) \) get closer and closer to \( L \) as \( x \) becomes closer to \( a \).
  • The limit tells us what value a function is approaching, not necessarily the value it actually reaches.
  • Limits help us to understand the behavior of functions at points where they might not even be defined.
When addressing continuity, limits are used to check if the function's trend aligns with its actual value at a specific point.
Function Values
The function value at a given point is simply the output of the function when we input that point. For instance, if \( f(x) = x^2 \), then \( f(4) = 4^2 = 16 \). Knowing function values helps to determine how a function behaves at specific points.
  • Function values are exact outputs, unlike limits, which deal with trends and behavior.
  • Finding \( f(a) \), where \( a \) is some number, is crucial when checking for continuity.
When proving continuity, the function value at the point of interest must align exactly with the limit of the function as it approaches that point. Ensuring this alignment is a core part of determining if a function is continuous.
Continuous Functions
A function is considered continuous at a point if there are no jumps, breaks, or holes at that point. Importantly, three conditions must be met for a function \( f \) to be considered continuous at \( x = a \):
  • The limit \( \lim_{{x \to a}} f(x) \) must exist.
  • The function value \( f(a) \) must be defined.
  • Both the limit and the function value at \( a \) must be equal; i.e., \( \lim_{{x \to a}} f(x) = f(a) \).
For example, in the context of this exercise, the function is continuous at \( x = 4 \) if \( \lim_{{x \to 4}} f(x) = f(4) \). This means as \( x \) nears 4, \( f(x) \) consistently approaches the same value as \( f(4) \), ensuring the function doesn't do anything unexpected at that point. Understanding these criteria will help you identify continuous functions accurately.

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Most popular questions from this chapter

Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. $$ f(x)=\left\\{\begin{array}{ll}{\frac{1}{x+2}} & {\text { if } x \neq-2} \\\ {1} & {\text { if } x=-2}\end{array} \quad a=-2\right. $$

(a) A tank contains \(5000 \mathrm{L}\) of pure water. Brine that cotains \(30 \mathrm{g}\) of salt per liter of water is pumped into the tank at a rate of \(25 \mathrm{L} / \mathrm{min}\). Show that the concentration of salt after \(t\) minutes (in grams per liter) is $$C(t)=\frac{30 t}{200+t}$$ (b) What happens to the concentration as \(t \rightarrow \infty ?\)

Use continuity to evaluate the limit. $$ \lim _{x \rightarrow \pi} \sin (x+\sin x) $$

Let \(P\) represent the percentage of a city's electrical power that is produced by solar panels \(t\) years after January \(1,2000 .\) (a) What does dP/dt represent in this context? (b) Interpret the statement $$ \left.\frac{d P}{d t}\right|_{t=2}=3.5 $$

Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. $$ f(x)=\left\\{\begin{array}{ll}{\frac{x^{2}-x}{x^{2}-1}} & {\text { if } x \neq 1} \\ {1} & {\text { if } x=1}\end{array} \quad a=1\right. $$
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