Problem 39 - Thomas Calculus

Question

Find the limits in Exercises \(37-48.\) $$\lim _{x \rightarrow 2^{-}} \frac{3}{x-2}$$

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Identify the Type of Limit

The limit in question is \( \lim_{x \to 2^{-}} \frac{3}{x-2} \). The notation \( x \to 2^{-} \) indicates that we are approaching 2 from the left, or from values less than 2. This kind of limit is called a left-hand limit.

Step 2: Analyze the Behavior Near the Point

Consider the function \( f(x) = \frac{3}{x-2} \). As \( x \) approaches 2 from the left, \( x - 2 \) approaches 0 from the negative side. Therefore, \( \frac{3}{x-2} \) becomes a division of a positive number by a very small negative number, which causes the expression to approach negative infinity.

Step 3: Conclude the Limit

Since as \( x \to 2^{-} \), the value of \( \frac{3}{x-2} \) decreases without bound (approaches negative infinity), we conclude that \( \lim_{x \to 2^{-}} \frac{3}{x-2} = -\infty \).

Key Concepts

Left-Hand Limit

In calculus, when we discuss limits, we often look at the behavior of a function as it approaches a certain point. A left-hand limit considers what happens to a function's output as the input approaches a specific value from the left-hand side. This means we're approaching from values that are slightly less than the point in question.

For example, in the provided exercise, the limit expression is \( \lim_{x \to 2^{-}} \frac{3}{x-2} \). The notation \( x \to 2^{-} \) clearly indicates that we're approaching 2 from the left. As such, we're examining what happens when \( x \) is just a bit less than 2.

Left-hand limits help us understand the behavior of functions in situations where they might not be well-defined exactly at the point we're examining. By looking from one side only, we get an isolated perspective which can be essential for understanding unique function behaviors such as jumps or asymptotes.

Negative Infinity

Negative infinity is a concept in calculus that describes the behavior of functions as they decrease without bound. Imagine numbers getting endlessly larger but in the negative direction.

In our exercise, as \( x \) approaches 2 from the left, the function \( \frac{3}{x-2} \) becomes a division of 3 by a small negative number, which results in larger negative values. These numbers grow more significant in the negative sense, indicating that the function is trending towards negative infinity.

When we conclude that a limit leads to negative infinity, it means that there is no finite number that the function approaches; instead, it keeps decreasing and never settles at an upper or lower numerical bound. Negative infinity is crucial in calculus for understanding functions that have no limit in the finite sense but show a distinct pattern in their decrease.

Asymptotic Behavior

Asymptotic behavior in a function refers to a trend where the function approaches a certain line but never actually touches or crosses it. This line is often called an asymptote. The function can get arbitrarily close to the line, but there will always be some distance, however small, between them.

In terms of our example with \( \frac{3}{x-2} \), as \( x \) nears 2 from the left, the function approaches negative infinity. The line \( x = 2 \) acts as a vertical asymptote. As \( x \) gets closer and closer to 2 from the left, the function values dive towards negative infinity, showing that asymptotic behavior.

Understanding asymptotes is essential as they inform us about points where functions have discontinuities or undefined behavior, offering insights into the limits and bounds of a function's growth or decay.