Problem 32 - Calculus of a Single Variable

Question

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

Step-by-Step Solution

Answer

The limit as \(x \rightarrow-\infty\) for the given function is 0.

Step 1: Identify behavior at infinity

Recognize that as \(x \rightarrow -\infty\), both \(x^{4} - 1\) and \(x^{3} - 1\) tend to infinity. Therefore the function's limit will be determined by the relative growth rates of the numerator and the denominator. Here, a method called 'dividing by the highest power of \(x\)' in the denominator is used.

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

Divide every term in the numerator and the denominator by \(x^{4}\), which is the highest power in denominator. \[\lim _{x \rightarrow-\infty} \frac{\frac{\sqrt{x^{4}-1}}{x^{4}}}{\frac{x^{3}-1}{x^{4}}}\] On simplification, it changes to:\[\lim _{x \rightarrow-\infty} \frac{\sqrt{1-\frac{1}{x^{4}}}}{\frac{1}{x}-\frac{1}{x^{4}}}\]

Step 3: New limit for simplified function

Notice now that as \(x \rightarrow -\infty\), all terms with \(x\) in the denominator tend to zero except for the \(\frac{1}{x}\) term in the denominator which tends to negative infinity. This leaves the limit in a determinable form:\[\lim _{x \rightarrow-\infty} \frac{\sqrt{1-\frac{1}{x^{4}}}}{\frac{1}{x}} = \lim _{x \rightarrow-\infty} \frac{\sqrt{1}}{\frac{1}{x}} = 0\]

Key Concepts

Asymptotic Behavior

In calculus, asymptotic behavior describes how a function behaves as its input approaches a specific value, often infinity or negative infinity. When exploring limits at infinity, we analyze how the function's value changes as the variable grows larger or smaller without bound. In the original exercise, we examined how the function behaves as \( x \) approaches \(-\infty\). This task involved assessing the function's asymptotic behavior, focusing on the dominant terms in the numerator and denominator that influence the limit.Asymptotic behavior helps us understand the overarching tendencies of functions:

• If a function has an asymptote, it approaches a particular line more and more closely but never actually reaches it.
• Rational functions often involve horizontal or vertical asymptotes influenced by the degrees of the polynomials involved.
• Analyzing asymptotic behavior simplifies complex expressions, especially when encountering infinite or undefined forms.

In our specific problem, dividing every term by the highest power of \( x \) allowed us to observe which components tend to zero and which dominate the behavior at \( x \to -\infty \). This simplification is a common technique in handling limits involving large values of \( x \).

Growth Rates in Functions

When examining functions, understanding growth rates aids in predicting how quickly the value of a function increases or decreases. Growth rates are vital when comparing polynomials, as the relative sizes of terms with large exponents substantially influence the function's behavior:

• A polynomial with a higher degree term in the numerator or denominator often dictates the function's growth rate and asymptotic behavior.
• When a function's terms involve powers of \( x \), identifying the terms with the largest exponent is crucial to understanding the growth rates.
• In our exercise, the term \( x^4 \) in the denominator greatly affected the function's growth as \( x \) approached \(-\infty\).

Recognizing growth rates assists in simplifying complicated fractions, especially when evaluating limits. By focusing on the largest exponent terms, we can make informed approximations about how the function behaves as \( x \) tends towards infinity or negative infinity, leading to more straightforward solutions.

Infinite Limits

In calculus, infinite limits explore the behavior of functions as they approach infinite or negative infinite input values. Unlike finite limits, which determine precise values, infinite limits help us characterize the growth or decay of functions over vast ranges of \( x \).Here are some key aspects of infinite limits:

• Infinite limits provide insights into whether functions grow without bound or stabilize into a particular trend.
• In our exercise, we assessed the function as \( x \to -\infty \), simplifying it to a form that reveals the limit approaches zero.
• Dividing terms by the highest degree of \( x \) often helps highlight dominant terms, making complex expressions manageable.

Understanding infinite limits is particularly important when dealing with indeterminate forms such as \( \frac{\infty}{\infty} \), as it allows for resolution into determinate forms or highlights asymptotic tendencies. By simplifying the function and focusing on infinite behavior, we find the overall limit despite the unbounded nature of the components when \( x \to -\infty \).