Problem 62 - Thomas Calculus

Question

Find the limits in Exercises \(59-62\). $$\lim \left(\frac{1}{x^{1 / 3}}-\frac{1}{(x-1)^{4 / 3}}\right)\mathrm{as}$$ $$\begin{array}{ll}{\text { a. }} & {x \rightarrow 0^{+}} & {\text { b. }} & {x \rightarrow 0^{-}} \\ {\text { c. }} & {x \rightarrow 1^{+}} & {\text { d. }} & {x \rightarrow 1^{-}}\end{array}$$

Step-by-Step Solution

Answer

a: infinity, b: -infinity, c: -infinity, d: infinity.

Step 1: Understanding the Expression

The given expression is \( \frac{1}{x^{1/3}} - \frac{1}{(x-1)^{4/3}} \). We'll need to evaluate this limit for different values of \( x \) approaching 0 and 1 from both positive and negative directions.

Step 2: Evaluating the Limit as x Approaches 0 from the Positive Direction

We need to find \( \lim_{{x \to 0^+}} \left(\frac{1}{x^{1/3}} - \frac{1}{(x-1)^{4/3}}\right) \).For \( x \to 0^+ \), \( x^{1/3} \) approaches 0, making \( \frac{1}{x^{1/3}} \) approach infinity. Meanwhile, \((x-1)^{4/3} \approx (-1)^{4/3} = -1\), so \( \frac{1}{(x-1)^{4/3}} \to -1\). Thus, the expression approaches infinity.

Step 3: Evaluating the Limit as x Approaches 0 from the Negative Direction

We need to find \( \lim_{{x \to 0^-}} \left(\frac{1}{x^{1/3}} - \frac{1}{(x-1)^{4/3}}\right) \).For \( x \to 0^- \), \( x^{1/3} \to 0 \) with the same sign as \( x \), so \( \frac{1}{x^{1/3}} \) approaches negative infinity. Meanwhile, as before, \( \frac{1}{(x-1)^{4/3}} \to -1\). Therefore, the expression approaches negative infinity.

Step 4: Evaluating the Limit as x Approaches 1 from the Positive Direction

We need to find \( \lim_{{x \to 1^+}} \left(\frac{1}{x^{1/3}} - \frac{1}{(x-1)^{4/3}}\right) \).As \( x \to 1^+ \), \( x^{1/3} \to 1 \) and \((x-1)^{4/3}\) approaches 0 from the positive side, making \( \frac{1}{(x-1)^{4/3}} \to +\infty \). The resulting expression is \( 1 - (+\infty) = -\infty \).

Step 5: Evaluating the Limit as x Approaches 1 from the Negative Direction

We need to find \( \lim_{{x \to 1^-}} \left(\frac{1}{x^{1/3}} - \frac{1}{(x-1)^{4/3}}\right) \).As \( x \to 1^- \), \( x^{1/3} \to 1 \) and \((x-1)^{4/3}\) approaches 0 from the negative side, making \( \frac{1}{(x-1)^{4/3}} \to -\infty \). The resulting expression is \( 1 - (-\infty) = +\infty \).

Key Concepts

Understanding Limits

When we talk about limits in calculus, we're essentially looking at what value a function approaches as the input gets closer to a specific number or point. It's like trying to figure out where a function is headed as we zoom in closer and closer to a spot on the number line.

Limits can exist in multiple dimensions, meaning we can approach a point from the left, from the right, or even in a more global sense. This can be particularly crucial when we deal with real-world problems where exact values may be difficult or impossible to obtain. For example, when you're evaluating a function at a point where it's not explicitly defined because of a division by zero or other such undefined regions, limits help us determine behavior.

Approaching Values from Positive and Negative Directions

When evaluating limits, it's important to recognize the direction from which you're approaching a given point. Approaching from the positive side, denoted as \( x \to a^+ \), means we're coming from larger numbers toward a smaller one. Conversely, \( x \to a^- \) indicates an approach from smaller numbers toward a larger point.

• In the exercise provided, for example, as \( x \to 0^+ \), the terms like \( \frac{1}{x^{1/3}} \) can behave quite differently than when approaching from \( x \to 0^- \).
• This discrepancy can lead to outcomes like \(+\infty\) or \(-\infty\), showing how a tiny shift in direction can result in a big change in value.

Understanding these nuances is essential, as it helps with grasping how the expression behaves around critical points, especially in complex functions with radicals and powers.

Evaluating Expressions with Powers and Radicals

Expressions that include powers and radicals often present unique challenges and intricacies, especially when considering limits. Radicals can convert positive expressions into negative results, and powers dictate how an expression grows or shrinks.

• For powers like \( x^{1/3} \), the cube root of \( x \), results differ based on whether \( x \) is positive or negative.
• Consider \((x-1)^{4/3}\) in the given exercise: raising a negative number to a fractional power can yield different behaviors depending on the context.

This requires careful analysis to predict accurately how each component of the expression behaves as \( x \) approaches the limit. By rigorously examining these components, you gain deeper insights into how the overall function behaves, making it easier to comprehend and solve complex calculus problems.