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A right-hand limit refers to the behavior of a function as the input values approach a specific point from the right side, or from values greater than the target point. For example, with the expression \( \lim _{x \rightarrow 3^{+}} \frac{x-3}{\sqrt{x^{2}-9}} \), we are approaching 3 from values just slightly more than 3
(like 3.1 or 3.01). In the solution we've analyzed, we examine the behavior of the expression when \( x = 3 + h \) where \( h \) approaches zero from the positive side (\( h \to 0^+ \)). This gives us a new expression: \( \frac{h}{\sqrt{6h + h^2}} \). When we simplify this, the crucial observation is that both the numerator \( h \) and the square root part contain small positive quantities as \( h \to 0^+ \). This influences the limit to approach zero, resulting in a right-hand limit of 0. This means that when coming from the right towards 3, the given expression's value shrinks to zero.

Contrary to the right-hand limit, the left-hand limit focuses on the behavior of a function as the input values approach a specific point from the left, or from values less than the target point. If we consider \( \lim _{x \rightarrow 3^{-}} \frac{x-3}{\sqrt{x^{2}-9}} \), this means we are examining how the function behaves as \( x \) approaches 3 from values just below 3
(such as 2.9 or 2.99).To analyze this, we set \( x = 3 - h \) with \( h \to 0^+ \), meaning \( h \) approaches zero from the positive side. This results in the expression \( \frac{-h}{\sqrt{-6h + h^2}} \). However, the significant issue arises because \( -6h + h^2 \) turns negative as \( h \) gets close to zero from the positive side, leading to a negative value under the square root.Since the square root of a negative number is not defined within the real numbers, the expression cannot be evaluated, and thus, the left-hand limit does not exist. This means that as we approach 3 from the left, the expression does not settle into a predictable, finite value.

When we say a limit does not exist, it means that as we approach a specific input value, the function does not settle into a single predictable value. In this case with \( \lim _{x \rightarrow 3^{-}} \frac{x-3}{\sqrt{x^{2}-9}} \), the limit does not exist due to the square root of a negative number issue discussed previously.This is an example where mathematical operations cannot produce a real, finite result as you approach the point from one side. A limit might not exist because:

• The function heads off to infinity.


• The function oscillates between widely differing values.


• Technical mathematical reasons, like the square root of negative numbers here.

In essence, it’s crucial to conclude whether a limit exists with careful analysis, as simply substituting in values can often lead to incorrect conclusions when limits approach undefined or undefined-like behavior.