Problem 47 - Thomas Calculus

Question

Find the limits in Exercises \(37-48\). $$\lim _{x \rightarrow 0} \frac{4}{x^{2 / 5}}$$

Step-by-Step Solution

Answer

The limit approaches infinity as \( x \) approaches 0 from the positive side.

Step 1: Identify the Limit Expression

The expression given is \( \lim _{x \rightarrow 0} \frac{4}{x^{2 / 5}} \). We need to determine the behavior of this function as \( x \) approaches 0.

Step 2: Rewrite the Function

The function can be rewritten as \( 4 \cdot x^{-2/5} \). This will help in analyzing the behavior as \( x \) approaches 0.

Step 3: Analyze the Exponent

The exponent \( -\frac{2}{5} \) indicates an inverse power. Specifically, \( x^{-2/5} = \frac{1}{x^{2/5}} \), which behaves like a very large number when \( x \) is closer to zero but positive.

Step 4: Consider the Direction of Approach

As \( x \rightarrow 0^{+} \), \( x^{2/5} \rightarrow 0 \), making \( 1/x^{2/5} \rightarrow \infty \). Thus, the expression \( 4 \cdot x^{-2/5} \rightarrow \infty \).

Step 5: Combine Insights for the Limit

Since \( x^{-2/5} \rightarrow \infty \) as \( x \rightarrow 0^{+} \), the overall expression \( \lim_{x \rightarrow 0} \frac{4}{x^{2/5}} \) yields \( \infty \) as \( x \rightarrow 0^{+} \). The limit is undefined if considering a two-sided approach because it diverges to \( +\infty \) from positive direction and \( -\infty \) from negative direction.

Key Concepts

Analysis of Exponents

When dealing with expressions involving exponents, especially in calculus, understanding how different powers affect the behavior of a function is crucial. In the given limit problem, the exponent is \(-\frac{2}{5}\). Let's break this down:
\( x^{-2/5} \) is equivalent to \( \frac{1}{x^{2/5}} \), which indicates an inverse power. Here's what this transformation means:

• A negative exponent signifies that the variable is in the denominator. This implies a reciprocal relationship.
• The positive fraction \( \frac{2}{5} \) indicates the root of power. It is a fifth root, meaning the smaller the value of \(x\), the larger the power effect.

This reciprocal and fractional relationship greatly influences how the function behaves as \(x\) approaches zero.

Behavior of Functions Near Zero

Understanding the behavior of functions as variables approach zero involves analyzing the function's response in this proximity.
For \( \lim _{x \rightarrow 0} \frac{4}{x^{2/5}} \), we're interested in how \( x^{2/5} \) behaves near zero:

• As \( x \) gets closer to zero from the positive side, \( x^{2/5} \) becomes smaller than 1, eventually approaching but never reaching zero.
• Since \( x^{2/5} \) is ever-decreasing as \( x \) gets smaller, its reciprocal \( \frac{1}{x^{2/5}} \) becomes larger.

This analysis shows that near zero, particularly on the positive side, the function grows without bound, or approaches infinity. However, if \( x \) approached zero from the negative side, you'd face a different scenario, possibly involving complex numbers if considering real roots at certain powers of \(x\).

One-Sided Limits

One-sided limits focus on the behavior of functions as the input approaches a point from one direction: either from the left or right.
In this limit exercise, calculating \( \lim_{x \rightarrow 0^{+}} \frac{4}{x^{2/5}} \):

• The expression diverges to \( \infty \), as already analyzed.
• From the left side, \( \lim_{x \rightarrow 0^{-}} \frac{4}{x^{2/5}} \) would approach complex or undefined results due to negative bases of a fractional power.

This demonstrates why specifying a direction (one-sided limit) is essential as it defines how we evaluate the function's behavior. Calculating each directional limit helps understand potential singularities or infinities which define more complex overall behavior at the point.