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Chapter 1: Problem 29

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow-\infty} \frac{x x^{2}}{x^{2}+3 x-1} $$

Short Answer

Expert verified
The limit is \( - \infty \).

Step by step solution

01

Simplify the expression

First, write the given limit and simplify the numerator. The given limit is \ \ \ \ \[ \lim _{x \rightarrow - \infty} \frac{x \cdot x^{2}}{x^{2}+3x-1} \], which simplifies to \[ \lim _{x \rightarrow -\infty} \frac{x^{3}}{x^{2}+3x-1} \].
02

Factor out the highest power of x from the denominator

Next, factor out the highest power of \( x \) in the denominator. The denominator \( x^{2} + 3x - 1 \) can be rewritten as \[ x^{2}(1 + \frac{3}{x} - \frac{1}{x^{2}}) \].
03

Divide the numerator and denominator by \( x^{2} \)

Divide both the numerator and the denominator by \( x^{2} \): \[ \lim_{x \rightarrow -\infty} \frac{x^{3}}{x^{2}(1 + \frac{3}{x} - \frac{1}{x^{2}})} = \lim_{x \rightarrow -\infty} \frac{x}{1 + \frac{3}{x} - \frac{1}{x^{2}}}. \]
04

Evaluate the limit as x approaches \( -\infty \)

As \( x \rightarrow -\infty \), \( \frac{3}{x} \) and \( \frac{1}{x^{2}} \) both approach 0. So the expression simplifies to \[ \lim_{x \rightarrow -\infty} \frac{x}{1 + 0 - 0} = \lim_{x \rightarrow -\infty} x. \]
05

Determine the behavior of the expression

Since the limit is \( \lim_{x \rightarrow -\infty} x \), as \( x \) approaches \( -\infty \), \( x \) also approaches \( -\infty \). Therefore, the limit is \ \[- \infty \].

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Key Concepts

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

Asymptotic Behavior
Asymptotic behavior in calculus describes how a function behaves as the input either becomes very large or very small. In this particular problem, we look at what happens when the variable approaches negative infinity (\(x \rightarrow -\infty\)). To understand asymptotic behavior:
  • Focus on the terms with the highest power in the numerator and denominator, as these terms dominate the behavior of the function.
  • Notice how minor terms diminish in influence as \(x\) grows large or small.
For the given problem, the dominant term in both the numerator and denominator is \(x^2\). Simplifying, we observe how the function behaves when \(x\) becomes very large negatively.
Evaluating Limits
Evaluating limits involves determining the value a function approaches as its input nears a particular point. Here, we specifically investigate the behavior as \(x\) approaches negative infinity. Key steps include:
  • Simplifying the function to recognize significant terms.
  • Factorizing accordingly to see the predominant behaviors.
For example, in our calculation:
\[ \lim_{x \rightarrow -\infty} \frac{x \cdot x^{2}}{x^{2}+3x-1} \]
After simplification, this turns to
\[ \lim_{x \rightarrow -\infty} \frac{x^{3}}{x^{2}+3x-1} \] This simplification makes evaluating the limit more clear-cut. We make further progress by factoring out the highest powers of \(x\) in the denominator, aiding easier manipulations.
Infinite Limits
Infinite limits happen when the function鈥檚 value grows without bound as the input nears a specific value. In cases where \(x\) approaches positive or negative infinity, detecting infinite limits involves:
  • Identifying terms in the function that drive the value towards infinity.
  • Consolidating contributions of various terms to understand overall behavior.
For our function, simplifying leads us to:
\[ \lim_{x \rightarrow -\infty} \frac{x}{1 + \frac{3}{x} - \frac{1}{x^{2}}} \]
As \(x\rightarrow -\infty\), minor terms like \(\frac{3}{x}\) and \(\frac{1}{x^2}\) approach zero, leaving us with \(\lim_{x \rightarrow -\infty} x\). Since \(x\) approaches negative infinity, the final answer confirms that the limit becomes \(-\infty\).
Factoring
Factoring is a mathematical technique used to simplify expressions by pulling out common factors. It plays a crucial role in evaluating limits by reducing the complexity of the function. Here鈥檚 how factoring helps:
  • By factoring out the highest power of \(x\) in the denominator, you make the dominant term clear.
  • It simplifies the limit expression, making it easier to evaluate.
Let鈥檚 see this in action:
From:
\[x^{2} + 3x - 1\]
we factor out \(x^2\) resulting in:
\[x^2 (1 + \frac{3}{x} - \frac{1}{x^2})\]
which simplifies our limit to:
\[ \lim_{x \rightarrow -\infty} \frac{x}{1 + \frac{3}{x} - \frac{1}{x^{2}}} \] Factoring out the powers of \(x\) lets us focus on the term of interest, and disregard those that diminish as \(x\) becomes large negatively.

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Most popular questions from this chapter

COST-EFFICIENT DESIGN A cable is to be run from a power plant on one side of a river 900 meters wide to a factory on the other side, 3,000 meters downstream. The cable will be run in a straight line from the power plant to some point \(P\) on the opposite bank and then along the bank to the factory. The cost of running the cable across the water is \(\$ 5\) per meter, while the cost over land is \(\$ 4\) per meter. Let \(x\) be the distance from \(P\) to the point directly across the river from the power plant. Express the cost of installing the cable as a function of \(x\).

Find equations for these lines: a. Slope 5 and \(y\) intercept \((0,-4)\) b. Slope \(-2\) and contains \((1,3)\) c. Contains \((5,4)\) and is parallel to \(2 x+y=3\)

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