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Chapter 2: Problem 1

Explain in your own words the meaning of each of the following. $$(a)\lim _{x \rightarrow \infty} f(x)=5$$ $$(b)\lim _{x \rightarrow-\infty} f(x)=3$$

Short Answer

Expert verified
As \( x \to \infty \), \( f(x) \to 5 \); as \( x \to -\infty \), \( f(x) \to 3 \).

Step by step solution

01

Understanding Limits

Limits describe the behavior of functions as the input approaches a particular value or infinity. They provide a way to understand how a function behaves at points that may not be directly accessible or where the function does not have a defined value.
02

Meaning of \( \lim _{x \rightarrow \infty} f(x)=5 \)

This expression means that as the input \( x \) of the function \( f(x) \) becomes very large (approaches infinity), the values of \( f(x) \) get closer and closer to 5. The function 'levels out' or approaches 5, but it does not necessarily ever reach 5.
03

Meaning of \( \lim _{x \rightarrow -\infty} f(x)=3 \)

This expression indicates that as \( x \) becomes very large in the negative direction (approaches negative infinity), the values of \( f(x) \) get nearer and nearer to 3. Like before, this describes the long-term behavior of \( f(x) \) as \( x \) moves far to the left on the number line.

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Key Concepts

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

Infinity in Calculus
Infinity in calculus involves understanding values that grow without bound in the either positive or negative direction. It is not a number but a concept used to describe unending growth or limitless size.
When we say a function approaches infinity, we describe a trend where values of the function keep getting larger and larger forever. For example, the expression \( x \to \infty \) means that the value of \( x \) is increasing without bound, moving towards an infinitely large value.
Similarly, \( x \to -\infty \) describes a scenario where \( x \) is decreasing continuously, moving towards an infinitely small or highly negative value.
  • Infinity isn't an endpoint, it's an idea of endlessness.
  • It is essential in understanding the behavior of mathematical functions over unlimited intervals.
  • It helps in analyzing limits, especially concerning convergence or divergence of a function.
Behavior of Functions
The behavior of functions refers to how functions act or change as they approach certain critical points, such as infinity or specific numbers. Understanding function behavior is crucial for predicting trends and interpreting mathematical models.
Functions can behave differently based on their type, such as polynomials, exponential, or trigonometric functions. Each one has distinct characteristics and limits, which describe what happens to function values as inputs grow larger or smaller.
This behavior is often analyzed through limits, which help us understand the tendencies of functions at critical or boundary points, even if the function doesn't quite "reach" those points.
  • Behavior is shown mainly through graphs, allowing a visual understanding.
  • It helps identify horizontal asymptotes, indicating where function values level out over time.
  • Understanding function behavior can identify potential holes or undefined regions within functions.
Limits at Infinity
Limits at infinity give us a way to examine the trend in the behavior of a function as its input grows indefinitely in size. This concept describes what values the function is approaching as the inputs become extremely large or extremely tiny.
When we articulate, \( \lim_{x \rightarrow \infty} f(x) = 5 \), we convey that as \( x \) grows positively without bound, the function \( f(x) \) tends to stabilize around the value 5. Similarly, \( \lim_{x \rightarrow -\infty} f(x) = 3 \) means \( f(x) \) is settling near 3 as \( x \) heads negatively towards infinity.
  • Limits at infinity help in understanding the end behavior of functions.
  • They can determine horizontal asymptotes in the function's graph.
  • These limits provide insight into the long term trends of a function, critical in calculus and real-world applications.

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Most popular questions from this chapter

The velocity \(v(t)\) of a falling raindrop at time \(t\) is modeled by the equation\(v(t)=v^{*}\left(1-e^{-g t / v^{*}}\right)\) where \(g\) is the acceleration due to gravity and \(v *\) is theterminal velocity of the raindrop.$$$$ (a) Find \(\lim _{t \rightarrow \infty} v(t)\) .$$$$ (b) For a large raindrop in moderate rainfall, a typical terminal velocity is 7.5 \(\mathrm{m} / \mathrm{s} .\) How long does it take for the velocity of such a raindrop to reach 99\(\%\) of its terminal velocity? (Take \(g=9.8 \mathrm{m} / \mathrm{s}^{2} . )\)

$$\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta+\tan \theta}$$

Given that $$\lim _{x \rightarrow 2} f(x)=4 \quad \lim _{x \rightarrow 2} g(x)=-2 \quad \lim _{x \rightarrow 2} h(x)=0$$ find the limits that exist. If the limit does not exist, explain why. \(\begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 2}[f(x)+5 g(x)]} & {\text { (b) } \lim _{x \rightarrow 2}[g(x)]^{3}} \\ {(\mathrm{c}) \lim _{x \rightarrow 2} \sqrt{f(x)}} & {\text { (d) } \lim _{x \rightarrow 2} \frac{3 f(x)}{g(x)}} \\ {\text { (e) } \lim _{x \rightarrow 2} \frac{g(x)}{h(x)}} & {\text { (f) } \lim _{x \rightarrow 2} \frac{g(x) h(x)}{f(x)}}\end{array}\)

Explain why the function is discontinuous at the given number \(a\) . Sketch the graph of the function. \(f(x)=\left\\{\begin{array}{ll}{\frac{2 x^{2}-5 x-3}{x-3}} & {\text { if } x \neq 3} \\ {6} & {\text { if } x=3}\end{array}\right. \quad a=3\)

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