Explain why lim x→3^+ √ x−3 / 2−x does not exist.

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1. Consider the limit \( \lim_{{x \to 3^+}} \frac{\sqrt{x-3}}{2-x} \). We need to analyze the behavior of the function as \( x \) approaches 3 from the right.
2. As \( x \to 3^+ \), the expression \( \sqrt{x-3} \) approaches \( \sqrt{0} = 0 \) because \( x-3 \) becomes a very small positive number.
3. Simultaneously, as \( x \to 3^+ \), the expression \( 2-x \) approaches \( 2-3 = -1 \), which is a negative number.
4. The fraction \( \frac{\sqrt{x-3}}{2-x} \) thus approaches \( \frac{0}{-1} = 0 \) from the positive side, but the denominator is negative, indicating a sign change.
5. Since the denominator approaches a negative value and the numerator approaches zero, the limit does not exist because the function approaches different values from different sides of 3.

Key Concepts

Limits

In calculus, a limit describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. The notation lim x→c f(x) indicates the limit of f(x) as x approaches c, which can be from the left (c^-) or the right (c^+).

One-Sided Limits

One-sided limits are limits that consider the behavior of a function as the input approaches a specific value from one side only. The right-hand limit, denoted as lim x→c^+ f(x), examines the function as x approaches c from values greater than c. If the one-sided limits do not match or do not exist, the overall limit at that point does not exist.

Undefined Expressions

An expression is considered undefined when it leads to a situation that cannot be resolved mathematically, such as division by zero. In the context of limits, if the function approaches a form like 0/0 or ∞/∞, it indicates that the limit may not exist. Understanding how to identify and analyze these forms is crucial for determining the existence of limits.