Problem 18
Question
Show that \(y=3-x^{2}\) is a curvilinear asymptote of the graph of \(f(x)=\left(2+3 x-x^{3}\right) / x .\) Sketch the graph of \(y=f(x)\) showing this asymptotic behavior.
Step-by-Step Solution
Verified Answer
The curvilinear asymptote is \(y=3-x^2\).
1Step 1: Express the Function
The function given is \( f(x) = \frac{2 + 3x - x^3}{x} \). This can be rewritten by performing polynomial long division, or by simply simplifying: \( f(x) = -x^2 + 3 + \frac{2}{x} \).
2Step 2: Identify Asymptotic Behavior at Infinity
As \( x \) approaches infinity, the term \( \frac{2}{x} \) approaches 0. Hence, for large values of \( x \), the function \( f(x) \) approximates \( -x^2 + 3 \). Comparing with the given asymptote \( y = 3 - x^2 \), it's clear that \( f(x) \) tends towards this expression.
3Step 3: Confirm Limiting Behavior at Infinity
Evaluate the limit \( \lim_{x \to \pm \infty} \left( f(x) - (3 - x^2) \right) \). Simplifying gives \( \lim_{x \to \pm \infty} \frac{2}{x} = 0 \). This confirms \( y = 3 - x^2 \) as a curvilinear asymptote.
4Step 4: Sketch the Graph
Draw the function \( y = f(x) = -x^2 + 3 + \frac{2}{x} \). Also sketch the asymptote line \( y = 3 - x^2 \). As \( x \) increases or decreases without bound, the graph of \( f(x) \) should approach the line \( y = 3 - x^2 \). Note critical points and any intercepts to ensure a clear graph representation.
Key Concepts
Curvilinear AsymptotesGraph SketchingLimits at Infinity
Curvilinear Asymptotes
Curvilinear asymptotes represent how a function behaves at extreme values of its variable, usually as the variable approaches infinity or negative infinity. Unlike traditional horizontal or vertical asymptotes, which are straight lines, curvilinear asymptotes do not have to be linear. In this context, we are dealing with the function \( f(x) = \frac{2 + 3x - x^3}{x} \). Upon simplifying, this becomes \( f(x) = -x^2 + 3 + \frac{2}{x} \). The curvilinear asymptote given is \( y = 3 - x^2 \).
Essentially, as \( x \) grows very large (either positively or negatively), the function \( f(x) \) behaves similarly to its asymptotic equation \( y = 3 - x^2 \). To confirm this, we look at the difference between \( f(x) \) and the asymptote: \( f(x) - (3 - x^2) = \frac{2}{x} \). As \( x \to \infty \) or \( x \to -\infty \), this term tends to zero, showing that \( f(x) \) closely approaches \( y = 3 - x^2 \) when \( x \) is very large or very small.
Essentially, as \( x \) grows very large (either positively or negatively), the function \( f(x) \) behaves similarly to its asymptotic equation \( y = 3 - x^2 \). To confirm this, we look at the difference between \( f(x) \) and the asymptote: \( f(x) - (3 - x^2) = \frac{2}{x} \). As \( x \to \infty \) or \( x \to -\infty \), this term tends to zero, showing that \( f(x) \) closely approaches \( y = 3 - x^2 \) when \( x \) is very large or very small.
Graph Sketching
Graph sketching is crucial for visually understanding the behavior of functions, particularly in relation to their asymptotic behavior. To sketch the graph of \( f(x) = -x^2 + 3 + \frac{2}{x} \), follow these simple steps:
- Identify the general shape dictated by \( -x^2 + 3 \), which is a downward-opening parabola shifted up by 3 units.
- Note the presence of the term \( \frac{2}{x} \). This term has a negligible effect for large values of \( x \), as it tends towards zero, but causes the graph to deviate slightly from the straight parabolic path for smaller \( x \).
- Compare this curve against its asymptote \( y = 3 - x^2 \). For large \( x \), the graphs will seem almost indistinguishable because \( \frac{2}{x} \) becomes insignificant.
- Add key features such as the vertex and intercepts to clarify the sketch. For example, the vertex occurs where \( x = 0 \), making \( f(x) = 3 \).
Limits at Infinity
Understanding limits at infinity is key to analyzing the behavior of functions as they extend indefinitely in either direction. For the function \( f(x) = -x^2 + 3 + \frac{2}{x} \), let’s explore its limit as \( x \to \infty \) and \( x \to -\infty \).
The focus is primarily on the term \( \frac{2}{x} \). As \( x \to \infty \) or \( x \to -\infty \), this term approaches zero because dividing by an increasingly large number makes the quotient smaller. Hence, the function \( f(x) \) simplifies to approximately \( -x^2 + 3 \) for extreme values of \( x \).
Evaluating the formal limit, we have \( \lim_{x \to \pm \infty} \left( f(x) - (3 - x^2) \right) = \lim_{x \to \pm \infty} \frac{2}{x} = 0 \). This demonstrates that at these limits, the function \( f(x) \) and its curvilinear asymptote \( y = 3 - x^2 \) are effectively the same, apart from an infinitesimally small difference.
The focus is primarily on the term \( \frac{2}{x} \). As \( x \to \infty \) or \( x \to -\infty \), this term approaches zero because dividing by an increasingly large number makes the quotient smaller. Hence, the function \( f(x) \) simplifies to approximately \( -x^2 + 3 \) for extreme values of \( x \).
Evaluating the formal limit, we have \( \lim_{x \to \pm \infty} \left( f(x) - (3 - x^2) \right) = \lim_{x \to \pm \infty} \frac{2}{x} = 0 \). This demonstrates that at these limits, the function \( f(x) \) and its curvilinear asymptote \( y = 3 - x^2 \) are effectively the same, apart from an infinitesimally small difference.
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