91Ó°ÊÓ

One-sided limits are a fascinating concept in calculus, where we explore what happens to a function as it approaches a particular point from one side only. In this exercise, the limit we are dealing with is a right-hand limit, denoted by the notation \(-3^+\). This tells us that we should examine the behavior of the function as the variable \(t\) gets infinitely close to \(-3\) from values greater than \(-3\). It is important to focus only on the direction of the approach:

• Right-hand limit (\(-3^+\)): The approach towards -3 comes from the right side, meaning slightly greater than -3.
• Left-hand limit (\(-3^-\)): The approach would come from the left, slightly less than -3.

By understanding which way the function approaches the point, we can better predict how it behaves. As we analyze one-sided limits, we often simplify expressions to reveal more about the function's behavior.

Rational functions are expressions that represent the ratio of two polynomials. In this exercise, the function is expressed as \( \frac{t^2 - 9}{t+3} \). When dealing with limits, especially one-sided limits, it's crucial to simplify rational functions whenever possible. Consider the following aspects of rational functions:

• They are defined everywhere their denominator is not zero.
• Simplifying them can often help remove discontinuities.
• Factoring, as shown in the solution, reveals hidden behavior.

In our example, the key move was to factor the numerator into \( (t - 3)(t + 3) \). This showed that the expression was reducible, leading to a cancellation of terms. Factoring rational functions helps us see beyond apparent restrictions and better understand limits. In \( \frac{t^2 - 9}{t+3} \), simplifying revealed a neat depiction of the function as \( t - 3 \) for \( t eq -3 \). This simplification step often transforms complex expressions into manageable forms for limit evaluation.

Simplifying expressions is a valuable skill when evaluating limits. It involves transforming a complex expression into a simpler and more understandable form. In our exercise, understanding that \( t^2 - 9 \) is equivalent to \( (t-3)(t+3) \) was essential. Why do we simplify?

• To remove indeterminate forms such as \( \frac{0}{0} \).
• To identify cancellations that reveal useful information.
• To make substitution straightforward and evaluation exact.

After simplifying \( \frac{(t-3)(t+3)}{t+3}\), we found it equivalent to \( t - 3 \), valid everywhere except where \( t = -3 \). This cleared any discontinuity at the point of interest and enabled easy substitution. Without the obstacle of a denominator problem, the substitution of \(-3\) into \( t-3\) gave us the clear limit value of \(-6\). Simplifying expressions is thus a key step to making complex calculus problems easier to solve.