Problem 4 Sketch the graph of an example o... [FREE SOLUTION]

Chapter 1: Problem 4

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

Short Answer

Expert verified

Function has vertical asymptotes at $x = -2$ and $x = 2$, and a horizontal asymptote at $y = 0$.

Step by step solution

Identify Asymptotic Behavior

The conditions given indicate that as $x$ approaches 2, the function $f(x)$ tends towards infinity, suggesting a vertical asymptote at $x = 2$. Similarly, as $x$ approaches $-2$ from the right ($x \rightarrow -2^{+}$), $f(x)$ goes to infinity, and from the left ($x \rightarrow -2^{-}$), $f(x)$ goes to negative infinity, suggesting another vertical asymptote at $x = -2$.

Determine End Behavior

The limits as $x$ approaches infinity and negative infinity ($\lim_{x \rightarrow \infty} f(x) = 0$ and $\lim_{x \rightarrow -\infty} f(x) = 0$) indicate that the function $f(x)$ approaches the horizontal axis at both ends, suggesting horizontal asymptotes at $y = 0$.

Find Intercept

The condition $f(0) = 0$ means that the graph of the function passes through the origin (0,0).

Sketch the Graph

Taking all these observations into account, a function that satisfies these conditions could look like a rational function with vertical asymptotes at $x = -2$ and $x = 2$, passing through the origin and approaching the x-axis as $x$ moves towards positive or negative infinity. One such example could be $f(x) = \frac{x(x+2)}{(x-2)^2}$.

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Key Concepts

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

Limits

Limits are fundamental in understanding how a function behaves as its inputs approach a certain value. In essence, limits help determine what value a function will approach as the input gets closer and closer to a specific number.
For example:

The notation $ \lim _{x \rightarrow 2} f(x)=\infty$ means that as $x$ approaches 2, the function value increases without bound. This tells us about how the function behaves near $x = 2$, rather than the actual value at $x = 2$.
Similarly, $ \lim _{x \rightarrow-2^{+}} f(x)=\infty $ indicates that approaching $-2$ from the right results in infinitely increasing function values.
The notation $ \lim _{x \rightarrow-2^{-}} f(x)=-\infty $ shows that as $x$ approaches $-2$ from the left, the function values decrease infinitely.

These descriptions enable us to predict the 'end behavior' of functions at the points where direct evaluation isn't possible, often due to division by zero or discontinuities.

Vertical Asymptotes

Vertical asymptotes represent values of $x$ where a function tends towards infinite increases or decreases. Unlike a function's roots, vertical asymptotes occur where the function shoots off to infinity within its neighborhood.
For our exercise,

A vertical asymptote at $x = 2$ occurs because $\lim _{x \rightarrow 2} f(x) = \infty$, indicating a rapid increase in function value as $x$ approaches 2.
Additionally, $x = -2$ constitutes another vertical asymptote. The dual limits of $ \lim _{x \rightarrow-2^{+}} f(x)=\infty $ and $ \lim _{x \rightarrow-2^{-}} f(x)=-\infty$ show a split tendency, approaching positive infinity and negative infinity from different sides.

Vertical asymptotes suggest a type of division by zero, where the denominator is zero, but the numerator isn't, like in rational functions. They are key in sketching realistic graphs of these functions, marking points where the function isn't defined.

Horizontal Asymptotes

Horizontal asymptotes are lines that the graph of a function approaches as $x$ goes to positive or negative infinity. They provide insight into the behavior of the function at extreme values of $x$.
In our problem:

The limits $ \lim _{x \rightarrow \infty} f(x) = 0 $ and $ \lim _{x \rightarrow -\infty} f(x) = 0$ indicate horizontal asymptotes at $y = 0$. This tells us that the function flattens out and approaches the x-axis as $x$ moves away from the origin.

Horizontal asymptotes typically appear in rational functions, where the degree of the polynomial in the numerator is less than or equal to the degree of the polynomial in the denominator. While a function can approach horizontal asymptotes, they may never actually intersect them beyond a certain point.

Function Behavior

Understanding the overall behavior of a function involves analyzing different features such as asymptotes, limits, intercepts, and other important points.

One critical aspect is identifying intercepts like $f(0)=0$. This intercept is a precise point where the graph crosses the y-axis.
Collectively analyzing both vertical and horizontal asymptotes provides a framework to predict the function's graph form, especially at extremes.

In rational functions, for example, these behaviors highlight potential discontinuities and growth trends.
This overarching insight into a function helps sketch its graph more accurately. The choice of an example like $f(x) = \frac{x(x+2)}{(x-2)^2}$ illustrates how these behaviors meet the given exercise conditions. The graph would show rapid changes around $x = -2$ and $x = 2$, approach the x-axis for large magnitudes of $x$, and pass smoothly through the origin.

91影视

Short Answer

Step by step solution

Identify Asymptotic Behavior

Determine End Behavior

Find Intercept

Sketch the Graph

Key Concepts

Limits

Vertical Asymptotes

Horizontal Asymptotes

Function Behavior

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