Problem 71 - Essential Calculus Early Transcendentals

Question

Find the limit. $$ \lim _{x \rightarrow 3^{+}} \ln \left(x^{2}-9\right) $$

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Identify the Function and the Direction

We are given the limit \( \lim _{x \rightarrow 3^{+}} \ln (x^{2} - 9) \). This means we need to focus on how the natural log function \( \ln(y) \) behaves as \( y = x^2 - 9 \) approaches zero from the positive side as \( x \) approaches 3 from the right.

Step 2: Substitute Specific Values and Analyze Behavior

As \( x \) approaches 3 from the right, \( x \) can be thought of as a number slightly greater than 3, say \( x = 3 + \epsilon \) where \( \epsilon \) is small and positive. This makes \( x^2 - 9 = (3 + \epsilon)^2 - 9 = 9 + 6\epsilon + \epsilon^2 - 9 = 6\epsilon + \epsilon^2 \), which approaches 0 from the positive side as \( \epsilon \) approaches 0.

Step 3: Recognize the Limit Behavior of the Logarithm Function

The function \( \ln(y) \) tends to \(-\infty\) as \( y \rightarrow 0^{+} \). Since \( x^2 - 9 \rightarrow 0^{+} \) when \( x \rightarrow 3^{+} \), the expression \( \ln(x^2 - 9) \rightarrow -\infty \).

Step 4: Write the Final Limit

Thus, the limit \( \lim _{x \rightarrow 3^{+}} \ln (x^{2} - 9) \) is equal to \(-\infty\).

Key Concepts

Understanding the Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special logarithmic function that uses the mathematical constant "\( e \)" as its base. This constant, \( e \),approximates 2.718281828 and appears frequently in calculus, owing to its natural properties.

Key points to understand about the natural logarithm:

• Domain: The function \( \ln(x) \) is defined only for positive values of \( x \).If \( x = 0 \), the natural logarithm is undefined, and if \( x < 0 \), it is not considered within real numbers.
• Properties: \( \ln(1) = 0 \) since any number to the power of 0 is 1.For values of \( x \) inside (0,1), the natural logarithm returns negative outputs,whereas for values of \( x > 1 \), it yields positive outputs.
• Behavior as \( x \rightarrow 0^{+} \): As \( x \) approaches 0 from the positive side (\( x \rightarrow 0^{+} \)),\( \ln(x) \) approaches \( -\infty \). This means that even though \( x \) doesn't become negative,the logarithm declines sharply.

Behavior of Functions and Limits

When analyzing limits involving functions, it is essential to observe howfunctions behave as their input values approach certain points. This process isintegral to understanding and evaluating limits, particularly when inputsapproach values that result in undefined or infinite outputs.

• Consider \( y = x^2 - 9 \): The behavior of this function around \( x \rightarrow 3^{+} \)requires us to think of \( x \) as a value slightly greater than 3.Plugging in this thought (\( x = 3 + \varepsilon \) where \( \varepsilon > 0 \)), we observe \( y = 6\varepsilon + \varepsilon^2 \), indicating how \( y \) hovers around zero > from above as \( \varepsilon \rightarrow 0 \).
• Thus, observing how \( y \rightarrow 0^{+} \) helps determine that the limit based onthis behavior trends towards negative infinity when plugged into the natural logarithm.

Understanding the behavior of functions during such limits reveals how one can predictthe forthcoming behavior of more complex expressions.

Limit Approaching from the Right

In calculus, when we talk about a limit approaching from the right, or \( x \rightarrow c^{+} \),we focus on how a function behaves as its input approaches a particular point \( c \) from values greater than \( c \).This means we are considering the path from the direction of higher numbers towards \( c \).

Important aspects of approaching limits from the right:

• The notation \( x \rightarrow 3^{+} \) in our exercise specifies that \( x \) is approaching 3 from the right,or from a slightly larger position than 3.
• Analyzing \( x \rightarrow 3^{+} \) helps in predicting the behavior of the expression \( x^2 - 9 \), which becomes slightly positive even as it tends toward zero.
• Understanding this directional approach is crucial for predicting behaviors like\( \ln(x^2 - 9) \rightarrow -\infty \) because the logarithm is being fed by values which are tending to zero from the positive side.

Grasping this concept can often unlock the solutions to more complex limits, highlighting the directional nature of calculus.