91Ó°ÊÓ

When we discuss limits in calculus, one-sided limits help us understand the behavior of functions as they approach a specific point from one direction. In this exercise, we are particularly interested in the limit from the left side, denoted as \( x \rightarrow 3^{-} \). This notation means that we consider values of \( x \) that get closer and closer to 3, but they never actually reach it, staying on the left-hand side of 3.

• Left-Hand Limit: Approaching from values less than the point.
• Right-Hand Limit: Approaching from values greater than the point.

The concept of one-sided limits is crucial for understanding how functions behave near points of interest, especially when they exhibit different behaviors from different directions. In our problem, the left-sided approach gives us insight into how the function behaves near the value 3 from the lower values of \( x \).

Analyzing the behavior of a limit involves looking at what happens to a function as it gets infinitely close to a point, while not necessarily reaching that point. This helps us understand whether the function approaches a finite number, positive infinity, negative infinity, or fails to have a limit at all. In our example, the function \( \frac{-1}{x-3} \) exhibits specific behavior as \( x \rightarrow 3^{-} \).

• As \( x \) gets very close to 3 from the left, \( x-3 \) becomes a very small negative number.
• This causes \( \frac{-1}{x-3} \) to grow larger and larger positively, heading towards \( +\infty \).

Understanding limit behavior in this context is about recognizing the nature of the denominator as it approaches zero, causing the fraction to approach infinity. This knowledge is vital for solving calculus problems involving asymptotes or when determining the continuity of a function.

The absolute value function \( |x| \) plays an essential role in limits where we are concerned about the distance from a particular point, regardless of direction. Particularly in limits, it allows us to model scenarios where the function's behavior depends on which side of a point we are approaching from. For \(|x-3|\) with the limit \( x \rightarrow 3^{-} \), knowing how to handle absolute values is crucial.

• When \( x<3 \), \( |x-3|=-(x-3) \) because we are on the negative side of 3.
• The absolute value ensures we have a non-negative distance from zero, even though \( x-3 \) itself might be negative.

Since the absolute value can alter the expression based on direction, it's key for calculating precise limits. It helps reveal the true behavior of functions near critical points and ensures accurate interpretation of function trends as \( x \) approaches a limit from a designated side.