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Chapter 2: Problem 9

\(5-10\) Sketch the graph of an example of a function f that satisfies all of the given conditions. $$ \begin{array}{l}{f(0)=3, \quad \lim _{x \rightarrow 0^{-}} f(x)=4, \quad \lim _{x \rightarrow 0^{+}} f(x)=2} \\ {\lim _{x \rightarrow-\infty} f(x)=-\infty, \quad \lim _{x \rightarrow 4^{-}} f(x)=-\infty, \quad \lim _{x \rightarrow 4^{+}} f(x)=\infty} \\ {\lim _{x \rightarrow \infty} f(x)=3}\end{array} $$

Short Answer

Expert verified
A piecewise graph shows the behavior with discontinuities at $x=0$ and $x=4$, with asymptotic and infinite tendencies as described.

Step by step solution

01

Initialize the Graph

Start by marking the key points on the x-axis based on the problem statement. Specifically, note points like $x=0$ and $x=4$. These are critical points where the behavior of the function changes.
02

Plot $f(0)=3$

Place a solid point on the graph at $(0, 3)$. This indicates that $f(x)$ is equal to 3 when $x$ is exactly zero.
03

Determine Discontinuity at $x=0$

Considering the limits: for $x$ approaching $0$ from the left, the value is 4, and from the right, it is 2. Represent this by open circles at points $(0, 4)$ and $(0, 2)$, signifying the limit values without an actual function value there.
04

Assess Behavior As $x \rightarrow -\infty$

For $x$ approaching negative infinity, draw the left part of the curve from negative y-values (heading towards negative infinity) as $x$ decreases.
05

Analyze the Discontinuity at $x=4$

At $x=4$, there is a vertical asymptote. As $x$ approaches $4$ from the left, the function tends toward negative infinity, and from the right, it tends toward positive infinity. Draw these trends accordingly.
06

Examine Behavior As $x \rightarrow \infty$

For $x$ growing towards positive infinity, $f(x)$ approaches 3. Draw the rightmost part of the curve approaching a horizontal asymptote at $y=3$.
07

Connect the Pieces

Connect the individual parts of the function graph where there are no restrictions to form a continuous graph that satisfies all given conditions.

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Key Concepts

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

Limits
In calculus, a limit is used to describe the value a function approaches as the input approaches some value. Evaluating limits is essential for understanding the behavior of functions near specific points. For the function given in the exercise, we observe different directional limits at important points like 0 and 4.
  • For x approaching 0, from the left, the function approaches 4, while from the right, the function approaches 2. This indicates a jump discontinuity at x=0.
  • As x approaches negative infinity, the function dies off to negative infinity, showing its downward slope on the left side of the graph.
  • At x=4, the function diverges as x approaches from either side, indicating a vertical asymptote.
  • Lastly, as x approaches positive infinity, the function settles at 3, suggesting a horizontal asymptote at y=3.
Understanding these limit behaviors helps in accurately sketching graphs and analyzing function behavior at critical points.
Continuity
Continuity is a property of functions where a function is seamless and unbroken, fulfilling three conditions: the function is defined at a point, the limit exists at that point, and the limit value equals the function's value. In this exercise, continuity is not observed at key points like 0 and 4:
  • At x=0, the limits from the left and right differ, creating a jump discontinuity. You see the function values mismatched at a single point, with f(0)=3 while limits differ at 4 and 2, based on the side from which x approaches.
  • At x=4, a vertical asymptote signifies infinite discontinuity, as highlighted by limits diverging negatively and positively approaching 4 from both sides.
Such discontinuities are crucial in shaping the function's graph, affecting where we can draw a continuous line and where breaks or asymptotic behavior occur.
Asymptotes
Asymptotes are lines that a graph can approach but never touch or cross. They provide vital insights into the behavior of a function at various limits, signaling where and how the function diverges.
  • The exercise describes a vertical asymptote at x=4. Vertical asymptotes occur where the function increases or decreases without bound, diverging to positive or negative infinity.
  • A horizontal asymptote is present at y=3 for this function, indicating where the function levels off as x approaches infinity.
Understanding asymptotes helps in recognizing the directions in which the function increases or decreases infinitely and stabilizes. In graph sketching, they determine precise regions where the function can remain within certain bounds.
Function Behavior
Analyzing a function's behavior provides insights into its nature across different domains, essential for graph sketching.
  • Consider how the function behaves for large negative x values: the function decreases to negative infinity, implying a downward trajectory on the graph.
  • For x approaching 0 from left and right, the distinct limits suggest a sudden jump, indicating a dual-behavior spot at this point.
  • Near x=4, the behavior drastically changes, with vertical asymptotic divergence indicating the function's steep rise and fall near this point.
  • As x approaches positive infinity, calming behavior emerges, leveling off at y=3, suggesting consistent horizontal behavior at large x values.
By tracking these behaviors, one can translate abstract functional expressions into clear visual representations on a graph, informing us about trends, rises, falls, and plateaus throughout the graph's domain.

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Most popular questions from this chapter

\(51-52\) Determine whether \(\mathrm{f}^{\prime}(0)\) exists. \(f(x)=\left\\{\begin{array}{ll}{x \sin \frac{1}{x}} & {\text { if } x \neq 0} \\\ {0} & {\text { if } x=0}\end{array}\right.\)

(a) Sketch the graph of the function \(f(x)=x|x|.\) (b) For what values of \(x\) is f differentiable? (c) Find a formula for \(f^{\prime}\) .'

The point \(P(1,0)\) lies on the curve \(y=\sin (10 \pi / x)\). $$\begin{array}{l}{\text { (a) If } Q \text { is the point }(x, \sin (10 \pi / x)) \text { , find the slope of the secant }} \\ {\text { line } P Q \text { (correct to four decimal places) for } x=2,1.5,1.4 \text { , }} \\\ {1.3,1.2,1.1,0.5,0.6,0.7,0.8, \text { and } 0.9 . \text { Do the slopes }} \\\ {\text { appear to be approaching a limit? }}\end{array}$$ $$\begin{array}{l}{\text { (b) Use a graph of the curve to explain why the slopes of the }} \\ {\text { secant lines in part (a) are not close to the slope of the tan- }} \\ {\text { gent line at } P .}\end{array}$$ $$\begin{array}{l}{\text { (c) By choosing appropriate secant lines, estimate the slope of }} \\ {\text { the tangent line at } \mathrm{P} .}\end{array}$$

\(31-36\) Each limit represents the derivative of some function \(f\) at some number a. State such an \(f\) and a in each case. $$\lim _{h \rightarrow 0} \frac{\sqrt[4]{16+h}-2}{h}$$

Zoom in toward the points \((1,0),(0,1),\) and \((-1,0)\) on the graph of the function \(g(x)=\left(x^{2}-1\right)^{2 / 3} .\) What do you notice? Account for what you see in terms of the differentiability of \(g .\)
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