Problem 12 - Calculus with Concepts in Calculus

Question

Determine the infinite limit. $$ \lim _{x \rightarrow 0^{+}} \frac{2}{x^{1 / 4}} $$

Step-by-Step Solution

Answer

The limit is \( \infty \).

Step 1: Understand the expression

The expression given is \( \lim _{x \rightarrow 0^{+}} \frac{2}{x^{1 / 4}} \). As \( x \) approaches 0 from the positive side (denoted by \( x \rightarrow 0^{+} \)), we need to evaluate how the expression behaves.

Step 2: Identify the behavior of the denominator

As \( x \rightarrow 0^{+} \), the denominator \( x^{1/4} \) approaches 0. Being a root, it remains non-negative, and as it gets closer to 0, \( x^{1/4} \) becomes very small.

Step 3: Evaluate the fraction as the denominator approaches 0

Since the numerator (2) is a constant and the denominator \( x^{1/4} \) becomes very small as \( x \rightarrow 0^{+} \), the entire fraction \( \frac{2}{x^{1/4}} \) becomes very large.

Step 4: Conclusion about the infinite limit

As \( x \rightarrow 0^{+} \), \( \frac{2}{x^{1/4}} \rightarrow \infty \). Thus, the limit is positive infinity.

Key Concepts

Limit Evaluation

In calculus, limit evaluation is a fundamental process where we assess what value a function approaches as the input (or variable) approaches a certain point. In our problem, we have the limit expression:

• \( \lim _{x \rightarrow 0^{+}} \frac{2}{x^{1 / 4}} \)

The notation \( x \rightarrow 0^{+} \) indicates that \( x \) is approaching 0 from the positive side, meaning only very small positive values are considered. Our task is to determine what \( \frac{2}{x^{1 / 4}} \) approaches as \( x \) gets closer to 0 from these small positive values.

Denominator Behavior

Understanding denominator behavior is crucial when dealing with fractions, especially in limits. Here, the denominator of our function is \( x^{1/4} \). As \( x \rightarrow 0^{+} \), we observe that:

• \( x^{1/4} \) is a fourth root, which means it is non-negative for all \( x \geq 0 \).
• As \( x \) gets closer to 0, \( x^{1/4} \) also becomes smaller and smaller, approaching 0.

This behavior indicates that the denominator is shrinking towards 0 as \( x \) approaches 0 from the right. This is a key aspect of understanding how the whole fraction behaves.

Fraction Behavior

When evaluating the behavior of a fraction in limits, the interplay between its numerator and denominator is essential. In our expression, \( \frac{2}{x^{1/4}} \), the numerator is constantly 2:

• The numerator remains unchanged (2).
• The denominator, \( x^{1/4} \), approaches 0.

As the denominator becomes very small, the value of the entire fraction grows considerably. This is because dividing by a smaller and smaller positive number results in a larger value. Therefore, in this case, the fraction heads towards infinity as the denominator approaches zero.

Positive Infinity

Positive infinity is a concept used to describe the behavior of variables that grow without bound. In the context of our limit, as \( x \rightarrow 0^{+} \):

• The denominator \( x^{1/4} \) becomes extremely small yet positive.
• The fraction \( \frac{2}{x^{1/4}} \), therefore, increases rapidly without bound.

This means the value of the limit goes toward positive infinity, denoted by \( \infty \). Understanding positive infinity helps clarify that although the function doesn't reach a numerical value, it expresses growth in a specific direction.

Calculus Problem Solving

Solving calculus problems involves systematic approaches to dissect functions and their limits. In this exercise:

• We begin by understanding the notation and limits involved.
• Next, exploring the behavior of components within the limit, such as the denominator and numerator separately.
• Finally, concluding based on the interactions of these components, and interpreting the limit results.

Using these methods helps simplify and solve complex calculus problems, making challenging concepts like infinite limits more approachable.