91Ó°ÊÓ

When evaluating limits in calculus, it's essential to understand how a function behaves as it approaches a specific point. In limit analysis, we are interested in the value that a function approaches as the input approaches a particular value, even if the function is not defined at that value. This process requires careful examination of the function's behavior from both directions surrounding the point of interest. Limit analysis often involves determining whether the limit is finite, infinite, or does not exist.

To find the limit of a function, we start by checking the function for any obvious discontinuities or undefined points, like division by zero. If the function is smooth and continuous around the point in question, we can often substitute the value directly into the function. However, if there is a discontinuity, further analysis is needed to understand the function's behavior and potentially apply limit-solving techniques, such as left-hand and right-hand limits.

When dealing with potential discontinuities or indeterminate points in limits, one-sided limits are indispensable. They offer a closer look into how a function behaves as it approaches a particular point from one specific direction, either from the left or the right.

One-sided limits are denoted as:

• The left-hand limit: \( \lim_{b \rightarrow a^-} f(b) \)
• The right-hand limit: \( \lim_{b \rightarrow a^+} f(b) \)

By evaluating the function as it approaches from each direction independently, we can determine if the overall limit exists. If both one-sided limits are equal at the point of interest, the two-sided limit exists, and it is equal to those one-sided limits.

However, if the one-sided limits are different, as seen in our example for \( b \rightarrow -3 \), where the function approaches negative infinity from the left and positive infinity from the right, the overall limit does not exist.

Indeterminate forms arise in calculus when limits present themselves in ambiguous forms, such as division by zero. These forms require special techniques for evaluation, as they do not yield obvious answers. Some common indeterminate forms include:

• \(\frac{0}{0}\)
• \(\frac{\infty}{\infty}\)
• \(0 \times \infty\)
• \(\infty - \infty\)

Dealing with indeterminate forms often necessitates using tools such as L'Hôpital's Rule or algebraic manipulation to resolve the ambiguity.

In our exercise, the indeterminate form arises when \( b = -3 \), causing the denominator to approach zero, leading to undefined behavior in the form \( \frac{-2}{0} \) as \( b \rightarrow -3 \) from either direction. Special attention to each side separately, as seen with one-sided limits, can reveal the limit's non-existence in cases like these, where the function diverges to infinity.