Problem 64 - Precalculus with Limits

Question

In Exercises 49-68, find the limit by direct substitution. $$ \lim_{x \to 3}\ \textrm{ln}\ x$$

Step-by-Step Solution

Answer

The limit of \( \lim_{x \to 3}\ \textrm{ln}\ x \) is approximately 1.0986.

Step 1: Understanding Direct Substitution

Direct substitution is a simple method used to find limits. As the term suggests, we substitute the limiting value (in this case, x = 3), directly into the function and compute the result.

Step 2: Apply Direct Substitution

Substitute \(x = 3\) into the function: \( \lim_{x \to 3}\ \textrm{ln}\ x = \textrm{ln}\ 3 \)

Step 3: Find the Value of Natural Logarithm

Now find the natural logarithm of 3. This can be done using a calculator. The natural logarithm of 3 is approximately 1.0986.

Key Concepts

Direct Substitution

Direct substitution is one of the simplest techniques used to evaluate limits in calculus. It involves replacing the variable in the function with the value it approaches. For instance, in the limit expression \( \lim_{x \to 3}\ \textrm{ln}\ x \), we substitute \( x \) with 3 directly into the function. This straightforward approach works well when the function is continuous at the point of interest. It's like plugging x = 3 into the function \( \textrm{ln}\ x \) to compute the result directly without any complex operations.

However, direct substitution doesn’t work if it leads to indeterminate forms like \( \frac{0}{0} \). In such cases, additional techniques such as factoring, simplification, or L'Hôpital's rule are needed. For our example, it works smoothly, giving us an approximate result for \( \ln 3 \).

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \). It’s commonly encountered in continuous growth processes, like compound interest and population growth.

The function \( \ln x \) is defined only for positive values of \( x \). As \( x \) gets closer to zero, \( \ln x \) tends toward negative infinity. Importantly, this function describes how much time is needed for the exponential growth to reach a certain level. Here, when evaluating \( \ln 3 \), we are simply determining how much time is needed to grow from 1 to 3 if the rate of growth is continuous at the rate \( e \). For calculations, this can be done using a scientific calculator or logarithm tables to find that \( \ln 3 \approx 1.0986 \).

Limit Evaluation

In calculus, limit evaluation is a crucial concept that helps understand the behavior of functions as they approach certain points. Evaluating limits enables us to find values that functions approach as the input gets close to a given number. There are different methods for evaluating limits:

• Direct Substitution: Replace the variable with the limit point directly into the function, as discussed earlier.
• Factoring: Simplify the expression by factoring, often used when encountering an indeterminate form.
• L'Hôpital's Rule: Used when dealing with indeterminate forms like \( \frac{0}{0} \), by taking the derivative of the numerator and denominator.

In our example, we used direct substitution to find \( \lim_{x \to 3}\ \textrm{ln}\ x \). Since natural logarithms are continuous in their domain, substituting \( x = 3 \) into \( \ln x \) was straightforward, yielding the approximate value 1.0986.

Always consider checking if the function is continuous at the limit point when deciding how to evaluate a limit. Using appropriate methods ensures accurate and efficient problem-solving.