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Chapter 2: Problem 79

In the following exercises, sketch the graph of a function with the given properties. $$\lim _{x \rightarrow-\infty} f(x)=2, \lim _{x \rightarrow-2} f(x)=-\infty$$ $$\lim _{x \rightarrow \infty} f(x)=2, f(0)=0$$

Short Answer

Expert verified
Sketch a curve approaching y=2 and x=-2 as asymptotes with a point at (0,0).

Step by step solution

01

Identify Asymptotic Behavior

Analyze the given limits to understand the overall shape and behavior of the function. As \( x \to -\infty \) and \( x \to \infty \), \( f(x) \to 2 \). This implies horizontal asymptotes of \( y = 2 \) for both ends of the function graph.
02

Consider the Vertical Asymptote

The limit \( \lim _{x \rightarrow -2} f(x)=-\infty \) indicates a vertical asymptote at \( x = -2 \). As \( x \) approaches \( -2 \) from either side, \( f(x) \) decreases to negative infinity.
03

Mark the Point Value

Identify the specific value of the function given by \( f(0) = 0 \). This gives a distinct point on the graph at \( (0,0) \).

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Key Concepts

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

Understanding the Limit of a Function
The limit of a function is a fundamental concept in calculus that describes how a function behaves as it approaches a specific point or infinity. When we say \( \lim_{x \to a} f(x) = L \), it means that as \( x \) gets closer and closer to \( a \), the value of \( f(x) \) gets closer to \( L \). This concept is crucial for understanding the long-term behavior of graphs.
In the given exercise:
  • As \( x \to -\infty \), \( f(x) \to 2 \), meaning the function levels off to 2 as \( x \) moves to the left past all finite numbers.
  • Similarly, as \( x \to \infty \), \( f(x) \to 2 \), which implies the same behavior on the right side of the graph.
  • The limit \( \lim_{x \to -2} f(x) = -\infty \) tells us that the function dives downwards indefinitely as it nears \( x = -2 \).
The presence of such limits can help us predict and sketch the overall behavior of the function and identify potential asymptotes.
Exploring Asymptotic Behavior
Asymptotic behavior describes how a function approaches a line, curve, or value as the input either goes to a specific point or tends to infinity. This behavior is visualized as the function getting extremely close to a line but never actually touching it.
  • Horizontal Asymptotes: For this function, as \( x \to \pm \infty \), \( f(x) \to 2 \). This means it has a horizontal asymptote at \( y = 2 \). The graph hugs this horizontal line as \( x \) goes both to the left and right extremes. Horizontal asymptotes indicate the level at which functions stabilize.
  • Vertical Asymptotes: The limit \( \lim_{x \to -2} f(x) = -\infty \) denotes a vertical asymptote at \( x = -2 \). Here, the function heads to negative infinity, indicating the graph suddenly drops down near this point.
Identifying these asymptotes is crucial when analyzing or sketching a function鈥檚 graph, as they guide the shape and boundary behaviors.
Mastering Graph Sketching
Graph sketching is an art that combines mathematical knowledge with visualization skills. It's the process of drawing a rough version of a graph based on its algebraic properties and behaviors.
Here鈥檚 how to approach graph sketching in the context of the given exercise:
  • Start by plotting known points. We know \( f(0) = 0 \), so the point \( (0,0) \) is on the graph.
  • Draw the asymptotes. Sketch the horizontal line \( y = 2 \) across the graph since it's the horizontal asymptote. Mark the vertical asymptote at \( x = -2 \) with a dashed line, indicating where the function sharply falls.
  • Fill in behavior near asymptotes and known points. As the function nears \( x = -\infty \) or \( x = \infty \), it should approach the line \( y = 2 \). Meanwhile, make sure the graph plunges down at \( x = -2 \).
With these elements in place, you can gain a solid understanding of how the function behaves and its key features visually.

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Most popular questions from this chapter

In the following exercises, use the precise definition of limit to prove the given limits. $$\lim _{x \rightarrow 2} \frac{2 x^{2}-3 x-2}{x-2}=5$$

In the following exercises, use the precise definition of limit to prove the given limits. $$\lim _{x \rightarrow 2}(5 x+8)=18$$

In the following exercises, use the precise definition of limit to prove the given infinite limits. $$\lim _{x \rightarrow-1} \frac{3}{(x+1)^{2}}=\infty$$

In the following exercises, find the value(s) of \(k\) that makes each function continuous over the given interval. $$ f(x)=\left\\{\begin{aligned} \frac{x^{2}+3 x+2}{x+2}, & x \neq-2 \\ k, & x=-2 \end{aligned}\right. $$

For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it? $$ \frac{2 x^{2}-5 x+3}{x-1} \text { at } x=1 $$
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