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Chapter 9: Problem 39

Find the limit. \(\lim _{x \rightarrow-\infty}\left(\frac{2 x}{x-1}+\frac{3 x}{x+1}\right)\)

Short Answer

Expert verified
The limit as x approaches negative infinity of \((\frac{2x}{x - 1} + \frac{3x}{x + 1})\) is 0

Step by step solution

01

Simplify the function

First, the function needs to be simplified. Rational functions can be added together by finding a common denominator. The common denominator in this case is \(x(x - 1)(x + 1)\). This gives: \(\frac{2x(x + 1) + 3x(x - 1)}{x(x - 1)(x + 1)} = \frac{2x^{2} + 2x + 3x^{2} - 3x}{x^{3} - x^{2} + x}\). This simplifies further to \(\frac{5x^{2} - x}{x^{3} - x^{2} + x}\)
02

Divide every term by \(x^{3}\)

To find the limit as x approaches negative infinity, divide every term in the rational function by \(x^{3}\) to get: \(\frac{(5x^{2}/x^{3}) - (x/x^{3})}{(x^{3}/x^{3}) - (x^{2}/x^{3}) + (x/x^{3})} = \frac{5/x - 1/x^{2}}{1 - 1/x + 1/x^{2}}\)
03

Apply the limit

This limit problem can now be solved by applying the concept that the limit as x approaches negative infinity of 1/x is 0. This gives: \(\lim _{x \rightarrow-\infty}\left(\frac{5/x - 1/x^{2}}{1 - 1/x + 1/x^{2}}\right) = \frac{5/(-\infty) - 1/(-\infty)^{2}}{1 - 1/(-\infty) + 1/(-\infty)^{2}} = \frac{0}{1} = 0\)

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Key Concepts

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

Limit of a Rational Function
Understanding the limit of a rational function is essential in calculus, especially when analyzing how functions behave as they tend toward infinity. A rational function is a ratio of two polynomials. To find the limit of such a function as the variable, say 'x', approaches a certain value, we need to look at the behavior of the numerator and the denominator separately.

When dealing with limits approaching infinity, one common technique is to divide each term by the highest power of 'x' found in the denominator. This usually simplifies the expression and reveals the function's end behavior. For example, the exercise presented involves a limit as 'x' approaches negative infinity. By dividing each term by the highest power of 'x', which is \(x^3\), the expression can be greatly simplified to show that as 'x' goes to negative infinity, the function will approach a certain value鈥攊n this case, zero.
Simplifying Rational Expressions
Rational expressions can often seem complex and difficult to work with. To simplify these expressions and make it easier to find their limits, we combine any like terms and factor wherever possible, essentially reducing the expression to its simplest form. This process involves finding the least common denominator (LCD) when combining fractions.

In the given exercise, the expression is simplified by first combining the two fractions with a common denominator. Once combined, this allows the cancelation of terms and further simplification. Simplifying rational expressions is a fundamental skill in calculus because it facilitates the analysis of functions' behaviors and making the calculation of limits more manageable.
Infinite Limits
In calculus, an infinite limit refers to the value that a function approaches as the variable approaches infinity, either positive or negative. When assessing infinite limits of rational functions, it's often useful to recognize that terms involving lower powers of 'x' become insignificant compared to those with higher powers of 'x' as 'x' grows larger in absolute value.

Applying this concept to the exercise at hand, as 'x' approaches negative infinity, the terms \(\frac{1}{x}\) and \(\frac{1}{x^2}\) approach zero, making them negligible. This often leads to a situation where the function's limit can be determined by examining the leading terms' coefficients or by recognizing that a term becomes insignificant in the limit process. Hence, as 'x' approaches negative infinity in the given function, the result can be determined to be zero.

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Most popular questions from this chapter

A manufacturer determines that the demand \(x\) for a product is inversely proportional to the square of the price \(p\). When the price is \(\$ 10\), the demand is 2500\. Find the revenue \(R\) as a function of \(x\) and approximate the change in revenue for a one-unit increase in sales when \(x=3000\). Make a sketch showing \(d R\) and \(\Delta R\).

The management of a company is considering three possible models for predicting the company's profits from 2003 through 2008 . Model I gives the expected annual profits if the current trends continue. Models II and III give the expected annual profits for various combinations of increased labor and energy costs. In each model, \(p\) is the profit (in billions of dollars) and \(t=0\) corresponds to 2003 . Model I: \(\quad p=0.03 t^{2}-0.01 t+3.39\) Model II: \(\quad p=0.08 t+3.36\) Model III: \(p=-0.07 t^{2}+0.05 t+3.38\) (a) Use a graphing utility to graph all three models in the same viewing window. (b) For which models are profits increasing during the interval from 2003 through 2008 ? (c) Which model is the most optimistic? Which is the most pessimistic? Which model would you choose? Explain.

Marginal Analysis, use differentials to approximate the change in cost, revenue, or profit corresponding to an increase in sales of one unit. For instance, in Exercise 29, approximate the change in cost as \(x\) increases from 12 units to 13 units. Then use a graphing utility to graph the function, and use the trace feature to verify your result. \(R=50 x-1.5 x^{2} \quad x=15\)

A business has a cost (in dollars) of \(C=0.5 x+500\) for producing \(x\) units. (a) Find the average cost function \(\bar{C}\). (b) Find \(\bar{C}\) when \(x=250\) and when \(x=1250\). (c) What is the limit of \(\bar{C}\) as \(x\) approaches infinity?

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=x+\frac{32}{x^{2}}\)
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