One Sided Limits

Let $g(x) = \frac{|x^2 + x - 6|}{x - 2}$

Find the limit as $x\rightarrow 2^{+} x\rightarrow 2^{-} x\rightarrow 2$

In this problem, you’re dealing with a one-sided limit, where you need to consider the behavior of the function as x approaches 2 from both the positive and negative directions. A one-sided limit means you examine the function's behavior either slightly greater than 2 or slightly less than 2, rather than from both directions at once. This is important for expressions involving absolute values, because the function might behave differently depending on which side of 2 you're approaching.

To solve it, you'd look at how the absolute value affects the expression on each side of 2. As you approach from the negative side, the terms inside the absolute value might change signs, while from the positive side, they might not. If the limit from the positive side and the limit from the negative side give different results, the overall limit does not exist in the normal sense, because for a limit to exist, both one-sided limits must be the same. If they differ, the function has no limit at that point.

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