Problem 32 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow 0^{+}} \sqrt{x}\left(1+\frac{1}{x^{2}}\right) $$

Step-by-Step Solution

Answer

The limit does not exist.

Step 1 - Understand the limit expression

The limit to evaluate is otag $$ otag otag otag otag otag otag otagotag otag otag otag otag otag otagotag otag otag otag otag otagotag otag otag otag otag otagotag otagotag otag otagotag otagotag otagotag$$otagotag otag otag. otagotag x otag x x \rightarrow 0^{+} x^{+} xx xx otag xx otagotag 0^+otag 0^+. otag otagotagotag goal otag notagnotag notagnotagotagotagnotagotag x root of steps

Step 2 - Simplify the expression inside the limit

Consider the expression inside the square root, otag $$ 1+otag otag otag otag. otag otag otag otag otag otagotagotag otag x^{2}. $$. As . Hence otag otag otag . Apply to the limit.

Step 3 Evaluate

Combine results otagotag otag otag otag. Result constant constant goes goes for infinity.

Key Concepts

One-Sided Limits

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.

Square Root Function

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

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.