91Ó°ÊÓ

One-sided limits are a fundamental concept in calculus. They help us understand the behavior of a function as it approaches a specific point from one direction, either from the left or the right. In the context of the original exercise, the notation \( \lim_{x \rightarrow 1^{+}} \) specifies a one-sided limit. The \( 1^{+} \) indicates that we are interested in the limit of \( x \) approaching 1 from the right side. This means \( x \) is slightly greater than 1 as it approaches 1.
The concept of one-sided limits allows us to study the continuity and behavior of functions in detail. This is particularly useful if the function behaves differently when approaching a point from different directions. "One-sided" here signifies that we are not considering what happens on both sides at once, but rather focusing on one direction.
Overall, understanding one-sided limits is important for analyzing functions where there could be a break in the graph, such as at a jump or vertical asymptote.

Infinite limits occur when the value of a function increases or decreases without bound as it approaches a certain point. In the given exercise, as \( x \) approaches 1 from the right, the value of \( \frac{1}{\sqrt{x-1}} \) becomes extremely large. This behavior describes what is called an infinite limit.
Since \( \sqrt{x-1} \to 0^{+} \), the denominator of the function approaches zero from the positive side. This makes the fraction \( \frac{1}{\sqrt{x-1}} \) grow larger, resulting in an infinite limit. When we say a function tends to infinity, it signifies that the outputs become indefinitely large as the input approaches the specified point.

• If the limit approaches positive infinity, we denote it as \( +\infty \).
• If it approaches negative infinity, we denote it as \( -\infty \).

Understanding infinite limits is essential for graphing and analyzing the asymptotic behavior of functions, particularly around vertical asymptotes.

Vertical asymptotes are specific lines on a graph where the function tends towards infinity. These occur when a function's limit approaches infinity from one or both sides as \( x \) approaches a certain value. In the context of the exercise, there is a vertical asymptote at \( x = 1 \) because \( \lim_{x \rightarrow 1^{+}} \frac{1}{\sqrt{x-1}} = +\infty \).
Key characteristics of vertical asymptotes include:

• The graph of the function will continue upwards or downwards without bounds near the asymptote, but will never actually touch or cross it.
• It usually occurs where a function has a division by zero in the limit.

Vertical asymptotes are crucial for understanding the behavior of rational functions and other expressions that contain terms with division, particularly as they approach undefined points. These asymptotes illustrate how functions behave similarly to a mathematical boundary, reflecting the limitations and constraints of certain functions.