91Ó°ÊÓ

When evaluating a limit, it is sometimes crucial to consider the direction from which we approach the point. This is known as one-sided limits.
In this exercise, we are asked to find the limit as \(x\) approaches 0 from the right, denoted as \(x \rightarrow 0^+\). Here's a brief overview:
- **Left-hand limit**: The limit as \(x\) approaches a point from the left, written as \(x \rightarrow a^-\).
- **Right-hand limit**: The limit as \(x\) approaches a point from the right, written as \(x \rightarrow a^+\).
For this particular limit, we are only concerned with values of \(x\) that are slightly greater than 0. This impacts our calculations since \( \frac{1}{x^2} \) becomes very large as \(x\) gets closer to 0 from the positive side.

The square root function, denoted as \( \sqrt{x} \), is fundamental in this limit problem.
Key properties include:

• \( \sqrt{x} \) is only defined for non-negative \(x\).
• As \(x\) approaches 0 from the right, \( \sqrt{x} \) also approaches 0.
• The function grows slower than linear functions as \(x\) increases.

In the limit \( \lim_\{x \rightarrow 0^{+}} \sqrt{x} \(1+\frac{1}{x^{2}}\) \), we need to understand how the square root and the expression inside it behave as \(x\) goes to 0 from the right.
Importantly, the term \( \frac{1}{x^{2}} \) becomes extremely large, overshadowing the +1. This means our function inside the square root approximates to \( \frac{1}{x^{2}} \) as \(x\) approaches 0 from the right.

Asymptotic behavior describes how a function behaves as its input grows very large or very small.
For the given limit, we specifically look at the behavior as \(x\) approaches 0 from the positive side. When \(x\) is very small but positive:
- \( \frac{1}{x^2} \) grows without bound, becoming extremely large.
The term inside the square root \(1+\frac{1}{x^2}\) is dominated by the \( \frac{1}{x^2} \) term.
Therefore, \( \sqrt{x(1+\frac{1}{x^2})} \) approximates to \( \sqrt{x \frac{1}{x^2}} \), simplifying further to \( \frac{1}{\sqrt{x}} \.\)
This behavior causes the entire expression to grow without bound as \(x\) approaches 0 from the right. Thus, the limit is \infty.