Problem 26 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(19-38,\) find the limit. $$ \lim _{x \rightarrow-\infty} \frac{x^{3}-4}{x^{2}+1} $$

Step-by-Step Solution

Answer

The limit of the function as x approaches -infinity is -infinity

Step 1: Rearrange Expression

The expression currently is \( \lim_{x \rightarrow-\infty} \frac{x^{3}-4}{x^{2}+1} \). Start by dividing every term in the expression by \(x^2\), the highest power of \(x\) in the denominator, to get: \[ \lim_{x \rightarrow-\infty} \frac{x^{3}/x^{2}-4/x^{2}}{x^{2}/x^{2}+1/x^{2}} \]It then simplifies to: \[ \lim_{x \rightarrow-\infty} \frac{x - 4/x^{2}}{1 + 1/x^{2}} \]

Step 2: Taking the Limit

Now that the expression is simplified, the limit as \(x\) approaches -infinity can be taken, since the values of \(4/x^{2}\) and \(1/x^{2}\) will approach 0, as \(x\) approaches -infinity. Then, that gives: \[ \lim_{x \rightarrow -\infty} x - \lim_{x \rightarrow -\infty} \frac{4}{x^{2}} | \lim_{x \rightarrow -\infty} 1 - \lim_{x \rightarrow -\infty} \frac{1}{x^{2}}\]Resulting in: \[ -\infty - 0 / 1 - 0\]

Step 3: Get final limit

The previous step simplifies to \(-\infty / 1\) which is equals to \(-\infty\). So, the limit of the given function as \(x\) approaches -infinity is \(-\infty\)

Key Concepts

Asymptotic behavior

In calculus, understanding the asymptotic behavior of a function is crucial when analyzing limits, especially when the variable approaches infinity or negative infinity. Asymptotic behavior refers to how a function behaves as the input values get very large in magnitude, whether positively or negatively. It helps us predict the function's tendencies without evaluating every single value.

For example, consider the function given in the exercise, \( \lim_{x \rightarrow-\infty} \frac{x^{3}-4}{x^{2}+1} \). Here, as \(x\) approaches \(-\infty\), the dominant term in the numerator, \(x^3\), outweighs the constant and all other terms, determining the function's overall behavior. Similarly, in the denominator, \(x^2\) mainly influences the output. This leads us to expect that the expression simplifies as \(-\infty\).

Understanding asymptotic behavior simplifies identifying the limits as it directs focus on the leading terms. This is especially beneficial when dealing with complex rational functions because it prevents unnecessary calculations and grounds the analysis in key mathematical principles.

Infinite limits

Infinite limits occur when the value of a function grows without bound as the input approaches a certain point or infinity. In our example, the notation \(\lim_{x \rightarrow-\infty}\) denotes that we are interested in the behavior of the function as \(x\) approaches negative infinity.

In the given exercise, as \(x\) tends towards \(-\infty\), the leading term \(x^3\) in the numerator of \(\frac{x^{3}-4}{x^{2}+1}\) causes the function to become increasingly large or negative. Since the term \(4/x^2\) in the numerator and \(1/x^2\) in the denominator approach zero, their contribution becomes negligible, thus reinforcing the result that the function approaches \(-\infty\).

By visualizing how these infinite limits work, you gain insight into why functions behave a certain way and can expect the function to be indefinitely large in a negative manner (\(-\infty\)) as \(x\) continues towards \(-\infty\). Infinite limits thus help provide a powerful yet straightforward understanding of a function's long-term behavior.

Rational functions

Rational functions are expressions that can be written as a fraction where both the numerator and the denominator are polynomials. They often appear in calculus because their asymptotic and limit behaviors can be quite revealing about the function's overall characteristics.

The exercise displays a typical rational function: \( \frac{x^{3}-4}{x^{2}+1} \). This illustrates key properties of rational functions -- as \(x\) approaches infinity or negative infinity, the behavior of the function is significantly determined by the leading terms of the numerator and denominator.

In rational functions, if the degree of the polynomial on top (numerator) is higher than the one on the bottom (denominator), the function tends to infinity or negative infinity. Conversely, if the degree is lower in the numerator, the function approaches zero. If they are equal, the function tends toward the ratio of their leading coefficients. Understanding these rules is vital for simplifying the process of finding limits and analyzing functions. Rational functions provide a structured pathway to navigate complex calculus problems.