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Limit evaluation is a crucial tool in calculus for understanding the behavior of functions as inputs approach specific values. In the given exercise, you're tasked with finding \(\lim_{x \rightarrow 1^-} \frac{1+x}{1-x}\). This expression requires analysis as it edges closer to the limit.
A limit evaluates what value a function approaches rather than what it equals at a particular point. In our exercise, substituting numbers like 0.9999, very close to 1, helps illustrate the behavior. Simultaneously, it avoids dividing by zero in scenarios where the denominator approaches zero.
For our fraction, substituting values approaching 1 from the left, such as 0.9999, yields increasingly larger numbers. This suggests that the function becomes infinitely large as \(x\) approaches 1. Proper understanding of evaluating limits can pinpoint if and how functions behave as they approach defined boundaries.

One-sided limits focus on evaluating functions from a specific direction: either from the left (denoted \(x \rightarrow c^-\)) or the right (\(x \rightarrow c^+\)). These are especially useful when functions behave differently approaching from each side.
In our case, we look at the left-hand limit \(\lim_{x \rightarrow 1^-} \frac{1+x}{1-x}\). As the approach is from the left, considering values just under 1 allows us to determine the function's behavior specifically from this direction.
Why are one-sided limits important? They help identify jumps, breaks, or approaches to infinity in functions. This specific exercise demonstrates that as you get closer to 1 from numbers smaller than 1, the function increases sharply, showcasing divergence to infinity on this side only.

Divergence in limits occurs when the value becomes unbounded, either positively or negatively. In terms of functions, this means the function does not settle toward a real or finite number but instead heads toward infinity.
In our example \(\lim_{x \rightarrow 1^-} \frac{1+x}{1-x}\), evaluating the expression as \(x\) inches closer to 1 results in the denominator approaching zero from a positive value. This progressively smaller denominator causes the fraction to grow without bound, indicating a divergence.
Recognizing divergence is crucial for real-world applications and theoretical functions. It identifies scenarios where predicted values might become unmanageable or undefined, impacting calculations and interpretations.