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In calculus, one-sided limits are an important tool to understand how functions behave as they approach a specific point from one direction. When we talk about a one-sided limit, we're referring to either a right-hand limit or a left-hand limit.

• The right-hand limit, denoted as \( \lim_{x \to a^+} f(x) \), describes the value that the function \( f(x) \) approaches as \( x \) approaches \( a \) from values greater than \( a \).
• On the other hand, the left-hand limit, displayed as \( \lim_{x \to a^-} f(x) \), identifies the value the function approaches as \( x \) moves toward \( a \) from values less than \( a \).

Understanding one-sided limits is crucial because sometimes, the behavior of a function differs depending on the direction from which \( x \) approaches the point. For instance, in the exercise provided, \( \lim_{x \to 1^+} \frac{x+2}{x+1} \) means we examine \( \frac{x+2}{x+1} \) as \( x \) approaches 1 from the positive, or right-hand, side. Grasping one-sided limits deepens our understanding of a function's behavior at points of discontinuity or sharp turns.

Continuity in a function is an essential aspect to determine the behavior of functions around certain points. A function is said to be continuous at a point if three conditions are satisfied:

• The function \( f(x) \) is defined at the point \( x = a \).
• The limit \( \lim_{x \to a} f(x) \) exists.
• The limit of the function as \( x \) approaches \( a \) equals the function's value at \( a \), i.e., \( \lim_{x \to a} f(x) = f(a) \).

When a function is continuous at a point, it means there are no gaps, jumps, or holes at that point, making it easier to calculate limits. In the provided exercise, when we substitute \( x = 1 \) into the function \( \frac{x+2}{x+1} \), the result \( \frac{3}{2} \) confirms there is no discontinuity at \( x = 1 \), as the function is well-defined and the denominator is not zero. This continuity means the one-sided limit from the right is straightforward to calculate and matches the function's value at \( x = 1 \).

The process of limit calculation allows us to predict a function's behavior near a particular point, even if a direct computation at that point might be tricky. For straightforward functions, if a direct substitution of the approaching value into the function doesn’t result in division by zero or indeterminate forms, the limit can be calculated directly.Here's how you generally calculate limits:

• If substituting the point into the function gives a real number without issues, then the limit equals this value.
• If you encounter an indeterminate form like \( \frac{0}{0} \), you might need to simplify the expression or use techniques like L'Hôpital's Rule, factoring, or trigonometric identities, depending on the problem.

In the given problem, substituting \( x = 1 \) directly into the function \( \frac{x+2}{x+1} \) yields \( \frac{3}{2} \), indicating that there were no complexities or complications at this point. Therefore, the one-sided limit as \( x \to 1^+ \) is quickly determined to be \( \frac{3}{2} \). This seamless calculation underscores the benefit of dealing with continuous functions. Limit calculations thus provide precise ways to establish function behavior close to points of interest.