Problem 27 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow-\infty} \frac{x^{4}+3 x^{2}-2 x+7}{x^{3}+x+1} $$

Step-by-Step Solution

Answer

The limit is \(-\infty\)

Step 1: Identify the highest power of x in the numerator and denominator

The given expression is \ \lim _{x \rightarrow-\infty} \frac{x^{4}+3 x^{2}-2 x+7}{x^{3}+x+1} \ The highest power of x in the numerator is \(x^4\) and in the denominator is \(x^3\).

Step 2: Divide both the numerator and denominator by the highest power of x in the denominator

To simplify the expression, divide both the numerator and the denominator by \(x^3\): \ \lim _{x \rightarrow-\infty} \frac{\frac{x^{4}+3 x^{2}-2 x+7}{x^3}}{\frac{x^{3}+x+1}{x^3}} = \lim _{x \rightarrow-\infty} \frac{x + \frac{3}{x} - \frac{2}{x^2} + \frac{7}{x^3}}{1 + \frac{1}{x^2}}

Step 3: Simplify the limit as \(x\) approaches \(-\infty\)

As \(x\) approaches \(-\infty\), the terms \(\frac{3}{x}\), \(\frac{2}{x^2}\), and \(\frac{7}{x^3}\) in the numerator approach 0, and \(\frac{1}{x^2}\) in the denominator also approaches 0. Thus, we have: \ \lim _{x \rightarrow-\infty} \frac{x + 0 - 0 + 0}{1 + 0} = \lim _{x \rightarrow-\infty} x

Step 4: Determine the behavior of the limit

As \(x\) approaches \(-\infty\), \(x\) itself approaches \(-\infty\). Therefore: \ \lim _{x \rightarrow-\infty} x = -\infty

Key Concepts

Infinite Limits

When we talk about infinite limits in calculus, we are exploring the behavior of a function as the input value either gets very large positively or very large negatively. In this exercise, the limit is analyzed as x approaches \(-\infty\).
If the function grows higher and higher without bound, we say the limit is \(+\infty\). Conversely, if the function gets lower and lower without end, we say the limit is \(-\infty\).
In this problem, we look at the trend of a rational function as x becomes very large in the negative direction.

Limit Simplification

To simplify limits, especially those involving polynomials, it helps to identify the highest power of x in both the numerator and the denominator.
This allows us to break down complex expressions for easier handling.
In the exercise, which involves \lim _{x \rightarrow-\infty} \frac{x^{4}+3 x^{2}-2 x+7}{x^{3}+x+1}\, the highest power of x in the numerator is \(x^4\) and in the denominator is \(x^3\).
We divide both by \(x^3\), the highest power of x in the denominator, to simplify the expression.
By doing this, the function becomes easier to analyze as x approaches \(-\fty\). We're left with simpler terms that approach zero or another discernible limit.

Behavior of Functions

Understanding the behavior of functions as x approaches extreme values (like \(-\infty\) or \(+\infty\)) is crucial.
In our example, simplification leads us to the expression \ \frac{x + \frac{3}{x} - \frac{2}{x^2} + \frac{7}{x^3}}{1 + \frac{1}{x^2}} \. Here, as x grows negatively large, each fractional term containing x in the denominator trends towards zero.
This insight allows us to reduce the expression further to just \( x \), then analyze its behavior.
Since we're looking at limits as x approaches \(-\infty\), we know that \ x \ itself dives towards \(-\infty\). This tells us the function behaves asymptotically in this manner.

Asymptotic Behavior

As we examine the trend closer, the term asymptotic behavior comes into play. This refers to how a function behaves as it approaches a particular line, known as an asymptote, but never really hits it.
In this case, as x approaches \(-\infty\), the simplified function \( x \) essentially describes the asymptote.
We conclude: the rational function here behaves asymptotically towards \(-\infty\) as x also trends towards \(-\infty\). Understanding such asymptotic behavior helps predict function trends for very large or very small inputs—a key concept in calculus.
By examining, simplifying, and understanding the behavior of the function, we effectively understand its infinite limit and overall trend as x reaches \(-\infty\).