Problem 35 - Calculus: Early Transcendentals

Question

Determine the infinite limit. \( \displaystyle \lim_{x \to 3^+}\ln (x^2 - 9) \)

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Understanding the Expression

We need to find the limit \( \lim_{x \to 3^+} \ln (x^2 - 9) \). This means finding the behavior of \( \ln(x^2 - 9) \) as \( x \) approaches 3 from the right.

Step 2: Identifying Problem Points

The expression inside the logarithm, \( x^2 - 9 \), simplifies to \( (x-3)(x+3) \). As \( x \to 3^+ \), \( x-3 \to 0^+ \) and \( x+3 \to 6 \). Thus, \( x^2 - 9 \) approaches 0 from the positive side.

Step 3: Rewriting the Logarithmic Expression

When \( x \to 3^+ \), we have \( \ln(x^2 - 9) = \ln((x-3)(x+3)) = \ln(x-3) + \ln(x+3) \). As \( x \to 3^+ \), \( \ln(x-3) \to -\infty \) because \( x-3 \to 0^+ \) and \( \ln(x+3) \to \ln 6 \) which is a finite constant.

Step 4: Determining the Limit

The dominant behavior is \( \ln(x-3) \to -\infty \). Therefore, the entire expression \( \ln((x-3)(x+3)) \to -\infty \).

Step 5: Conclusion

Hence, \( \lim_{x \to 3^+} \ln(x^2 - 9) = -\infty \). The function approaches negative infinity as \( x \) approaches 3 from the right.

Key Concepts

Logarithmic Functions

Logarithmic functions are a fundamental concept in mathematics, closely related to exponential functions. At its core, a logarithm answers the question: to what exponent must a base be raised to yield a certain number? For example, in the expression \( \log_{b} a = c \),\( b^c = a \). One of the most common bases used is the natural logarithm, denoted as \( \ln \), which uses the base \( e \), where \( e \approx 2.718 \).

Logarithmic functions have several key properties that are useful in calculus:

• \( \log_b(xy) = \log_b x + \log_b y \)
• \( \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y \)
• \( \log_b x^n = n \cdot \log_b x \)

These properties allow us to simplify and evaluate logarithmic expressions more easily.

Behavior at Vertical Asymptotes

A vertical asymptote occurs in mathematics when a function approaches a given value and makes a vertical line, such as when the input of a function results in undefined behavior. It is often characterized by an input value making the denominator of a fraction zero or causing some other undefined calculation. For logarithmic functions like \( \ln(x) \), vertical asymptotes appear where the argument of the logarithm approaches zero.

When analyzing a function near its vertical asymptote, pay attention to whether the function approaches infinity (positive or negative). In the exercise, as \( x \to 3^+ \), \( x^2 - 9 \) approaches 0 from the positive side, which causes \( \ln(x^2 - 9) \) to head towards negative infinity, unveiling a dominant component in the operation of logarithms.

Limits in Calculus

Limits are essential in calculus as they evaluate the behavior of functions as the input values approach a particular number. Rather than defining a function's value at certain points, limits describe what happens to the values of the function nearby.

For instance, the limit \( \lim_{x \to 3^+} \ln(x^2 - 9) \) investigates how \( \ln(x^2 - 9) \) behaves as \( x \) approaches 3 from the right side. This limit reveals that \( \ln(x^2 - 9) \) approaches \(-\infty\). Understanding limits, particularly infinite limits like this, is crucial to mastering calculus concepts and recognizing trends in function behavior.

Approaching Infinity

Approaching infinity in mathematical terms means that as the independent variable increases or decreases without bound, the function's value also increases or decreases without bound. It is a concept often seen in the context of limits, asymptotic behavior, and end-behavior analysis.

In the exercise, the function \( \ln(x^2 - 9) \) approaches negative infinity as \( x \to 3^+ \). This tells us that the logarithm of the values inside the parenthesis becomes extremely large in magnitude, but negative, as \( x^2 - 9 \) gets closer to zero. Understanding the nuances of approaching infinity is valuable in calculus, enabling students to predict end behavior and analyze infinite trends in functions.