Problem 24 - Thomas Calculus

Question

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of \(x :\) Divide numerator and denominator by the highest power of \(x\) in the denominator and proceed from there. Find the limits in Exercises \(23-36\). $$\lim _{x \rightarrow-\infty}\left(\frac{x^{2}+x-1}{8 x^{2}-3}\right)^{1 / 3}$$

Step-by-Step Solution

Answer

The limit is \(\frac{1}{2}\).

Step 1: Identify the Highest Power in Denominator

The given function is \( \left(\frac{x^{2}+x-1}{8x^{2}-3}\right)^{1/3} \). The highest power of \( x \) in the denominator is \( x^2 \).

Step 2: Divide Numerator and Denominator

Divide both the numerator and the denominator inside the function by \( x^2 \), the highest power in the denominator. This gives:\[\frac{x^2+ x - 1}{8x^2 - 3} = \frac{\frac{x^2}{x^2} + \frac{x}{x^2} - \frac{1}{x^2}}{\frac{8x^2}{x^2} - \frac{3}{x^2}} = \frac{1 + \frac{1}{x} - \frac{1}{x^2}}{8 - \frac{3}{x^2}}.\]

Step 3: Simplify the Expression Inside Limit

As \( x \rightarrow -\infty \), the terms with \( \frac{1}{x} \) and \( \frac{1}{x^2} \) tend towards zero. Thus, the expression simplifies to:\[\frac{1 + 0 - 0}{8 - 0} = \frac{1}{8}.\]

Step 4: Apply the Limit

Since we have the simplified fraction \( \frac{1}{8} \), the cube root of this value is considered as follows: The given limit is \( \lim_{x \rightarrow -\infty}\left( \frac{1}{8} \right)^{1/3} = \left( \frac{1}{8} \right)^{1/3}. \)

Step 5: Calculate the Cube Root

Find and express the cube root of \( \frac{1}{8} \):\[\left( \frac{1}{8} \right)^{1/3} = \frac{1}{2}.\]

Key Concepts

Noninteger Powers

When dealing with rational functions, it's common to encounter noninteger powers such as \(x^{1/3}\). Noninteger powers refer to roots of a number or variable. This means if you have a function like \(f(x) = (x^2)^{1/3}\), you're effectively looking at the cube root of \(x^2\). Understanding how to handle these powers is critical when calculating limits.

• Cube Roots: To simplify a cube root, think of it as finding a value that, when multiplied by itself three times, will give you the original number or expression inside the root. \(a^{1/3}\) will be a number that satisfies \(b^3 = a\).
• Computations: In our exercise, the final simplification resulted in finding the cube root of \(\frac{1}{8}\), which equaled \(\frac{1}{2}\). This was explained by realizing \(\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8}\).

Recognizing and working with noninteger powers involves understanding how they change as variables approach different limits, such as positive or negative infinity.

Negative Powers

Negative powers can seem a bit tricky at first, but they can be easily understood by recognizing their relationship with positive powers. A negative power of a number, such as \(x^{-2}\), is the reciprocal of the number raised to the positive power.

• Reciprocal Relationship: \(x^{-2}\) is the same as \(\frac{1}{x^{2}}\). This means if you have \(x^{-n}\), it equals \(\frac{1}{x^{n}}\).
• Application in Limits: In the exercise, dividing terms like \(\frac{1}{x}\) and \(\frac{1}{x^2}\) appear as \(x^{-1}\) and \(x^{-2}\). When \(x\) approaches negative infinity, these terms approach zero because dividing by an increasingly larger magnitude results in values close to zero.

Thus, negative powers are crucial for simplifying expressions, especially when infinitesimal smallness plays a role in limits.

Simplification of Expressions

Simplifying expressions is a key skill when you work with limits of rational functions, especially those involving complicated expressions with powers.

• Dividing by Highest Power: The important first step is to identify and divide the entire fraction by the highest power of \(x\) in the denominator. This makes every term inside the fraction less complex, as seen in our step-by-step solution where \(\frac{x^2 + x - 1}{8x^2 - 3}\) was divided by \(x^2\).
• Zero Limit Terms: After simplification, terms with negative exponents, like \(\frac{1}{x}\) and \(\frac{1}{x^2}\), tend to zero as \(x\) approaches plus or minus infinity. This greatly simplifies the expression and makes finding the limit straightforward.

The process of simplification allows us to transform a complex rational function into a simpler form, easing our way to calculate the corresponding limits efficiently.