The given problem involves evaluating the limit:

$\lim_{x \to 1} \left( \frac{3x}{x - 1} - \frac{3}{\ln(x)} \right).$

Let’s analyze this step-by-step.

Step 1: Check for indeterminate forms

When $x \to 1$, both terms $\frac{3x}{x - 1}$ and $\frac{3}{\ln(x)}$ can lead to indeterminate expressions. Specifically:

• For $\frac{3x}{x - 1}$: When $x \to 1$, this becomes $\frac{3 \cdot 1}{1 - 1} = \frac{3}{0}$, which suggests potential divergence or a more subtle behavior.
• For $\frac{3}{\ln(x)}$: As $x \to 1$, $\ln(1) = 0$, so this also takes the form $\frac{3}{0}$.

Both terms suggest that they individually blow up, so we need to analyze their combined behavior carefully.

Step 2: Simplify using series expansions

To handle limits where $x \to 1$, we can apply Taylor expansions around $x = 1$.

• For $x - 1$ near 0, we approximate:
  $x \approx 1 + (x - 1)$.

• For $\ln(x)$ near 1, the Taylor expansion gives:
  $\ln(x) \approx (x - 1) - \frac{(x - 1)^2}{2} + \cdots$.
  So, $\ln(x) \approx (x - 1)$ when higher-order terms are neglected.

Step 3: Substitute the expansions into the limit

1. First term:
   $\frac{3x}{x - 1} = \frac{3(1 + (x - 1))}{x - 1} = \frac{3 + 3(x - 1)}{x - 1} = 3 + 3.$

2. Second term:
   $\frac{3}{\ln(x)} \approx \frac{3}{x - 1}.$

Step 4: Combine the terms

The expression inside the limit becomes:

$\frac{3x}{x - 1} - \frac{3}{\ln(x)} \approx (3 + 3) - 3 = 3.$

Step 5: Evaluate the limit

Since the expression simplifies to a constant 3, the limit is:

$\boxed{3}.$

Would you like further clarifications or additional details?

Here are 5 related questions:

1. How do Taylor expansions help in limit calculations?
2. What are some other indeterminate forms besides $\frac{0}{0}$?
3. How can L'Hôpital's rule be applied to similar limits?
4. What are the conditions for applying Taylor series approximations?
5. What other techniques can handle limits with logarithms?

Tip: When limits result in indeterminate forms involving $0/0$ or $\infty - \infty$, Taylor expansions or L'Hôpital's rule are often effective strategies.