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Magazine Chapter 2: Problem 12
Determine the infinite limit. $$ \lim _{x \rightarrow 0^{+}} \frac{2}{x^{1 / 4}} $$
Short Answer
Expert verified
The limit is \( \infty \).
Step by step solution
01
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.
02
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.
03
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.
04
Conclusion about the infinite limit
As \( x \rightarrow 0^{+} \), \( \frac{2}{x^{1/4}} \rightarrow \infty \). Thus, the limit is positive infinity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
- \( \lim _{x \rightarrow 0^{+}} \frac{2}{x^{1 / 4}} \)
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.
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.
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.
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.
