Problem 26 - Calculus Early Transcendentals

Question

Determine the infinite limit. $$\lim _{x \rightarrow-3^{-}} \frac{x+2}{x+3}$$

Step-by-Step Solution

Answer

The limit is \(+\infty\).

Step 1: Understand the Problem

We need to find the limit of the function \( f(x) = \frac{x+2}{x+3} \) as \( x \) approaches \( -3 \) from the left. This means \( x \) is slightly less than \(-3\).

Step 2: Substitute the Approach Value

Directly substituting \( x = -3 \) gives \( \frac{-3+2}{-3+3} = \frac{-1}{0} \), indicating a division by zero, which suggests an infinite limit.

Step 3: Analyze the Behavior Near \(-3\)

For \( x \to -3^- \), \( x \) is slightly less than \(-3\). The denominator \( x+3 \) becomes a small negative number. The numerator \( x+2 \) is \(-1\) when \( x = -3 \).

Step 4: Determine the Direction of the Limit

Since the numerator approaches \(-1\) and the denominator approaches \(0^-\) (a very small negative number), the fraction \( \frac{x+2}{x+3} \) will approach positive infinity.

Key Concepts

Left-Hand Limits

The concept of left-hand limits involves determining the behavior of a function as the input values approach a specific point from the left side, denoted as \( x \to a^- \). In the original exercise, we examine the function \( f(x) = \frac{x+2}{x+3} \) as \( x \) approaches \(-3\) from the left.
To understand left-hand limits, imagine you are on a number line and moving towards the point \(-3\) from lesser values, such as \(-3.5, \ -3.1, \ -3.01\), etc.

• It's crucial to observe the values of the function as you approach the specified point—not just what the function equals AT that point.
• Rather than calculating the exact value when \( x = -3 \), we assess how the function behaves as it gets very close from the left.

In this problem, you can't directly evaluate the function by plugging in the approaching value because that results in division by zero. This is why understanding left-hand limits is invaluable—it allows us to predict the function's tendency or direction near the point of interest without needing the actual point to exist in the function's domain.

Division by Zero

Division by zero is a key concept that often arises in calculus, especially in limit problems. It occurs when an expression has a denominator of zero, which is mathematically undefined and resonates with infinite tendencies.
In the step-by-step solution, when we substitute \( x = -3 \) into the function \( \frac{x+2}{x+3} \), we obtain \( \frac{-1}{0} \). This highlights a division by zero scenario. It's imperative to understand that:

• When division by zero happens in a limit context, it signals a potential infinite limit.
• We shouldn't declare the expression as simply undefined; instead, we examine the sign and behavior of the numerator and denominator near the point of division by zero.

For limits, the behavior of the numerator and denominator as they approach zero helps us decide if the result tends towards positive or negative infinity, or if it oscillates between values. Thus, we can harness division by zero properly to uncover infinite limit tendencies.

Limits Approaching Infinity

When limits approach infinity, this means that as the input of a function nears a certain value, the output grows without bound. This concept is crucial when analyzing functions around points that cause division by zero, as illustrated in the original solution.
In the exercise, as \( x \to -3^− \), the expression \( \frac{x+2}{x+3} \) approaches positive infinity because:

• The numerator \(x+2\) will trend towards \(-1\).
• The denominator \(x+3\) will trend to \(0^−\), a slight negative number.

The combination of a constant numerator and a tiny negative denominator results in a large positive value for the fraction. The important takeaway is:

• When the numerator is negative and the denominator is a tiny negative, the overall fraction turns positive.
• If either the numerator or denominator sign varied, it would change the direction of infinity.

Understanding these tendencies helps in interpreting the direction of limits as they approach infinity in different scenarios, providing a deeper insight into the behavior of rational functions near vertical asymptotes.