𝑔(𝑥) = ((|x-1|)/(x-1))*(ln((x+2)/x^3))Find the values of at which has a vertical asymptote. 

To find the values of \( x \) at which the function \( g(x) = \frac{|x-1|}{x-1} \cdot \ln\left(\frac{x+2}{x^3}\right) \) has vertical asymptotes, we need to consider the points where the function becomes undefined or where the logarithmic expression approaches negative infinity (since the logarithm of zero or a negative number is undefined).

### Step 1: Analyze the expression \(\frac{|x-1|}{x-1}\)

The expression \(\frac{|x-1|}{x-1}\) can be broken down into two cases:

1. **Case 1: \( x > 1 \)**
   - \( |x-1| = x-1 \), so \(\frac{|x-1|}{x-1} = 1\).

2. **Case 2: \( x < 1 \)**
   - \( |x-1| = -(x-1) \), so \(\frac{|x-1|}{x-1} = -1\).

3. **Special Point: \( x = 1 \)**
   - At \( x = 1 \), the expression becomes \( \frac{0}{0} \), which is undefined. Thus, \( x = 1 \) is a potential location for a vertical asymptote.

### Step 2: Analyze the logarithmic expression \(\ln\left(\frac{x+2}{x^3}\right)\)

To determine where the logarithm might contribute to a vertical asymptote, we need to examine where the argument of the logarithm, \( \frac{x+2}{x^3} \), could be zero or undefined.

- **Undefined Points:** The expression \( \frac{x+2}{x^3} \) is undefined at \( x = 0 \) because it results in division by zero. Therefore, \( x = 0 \) is another potential location for a vertical asymptote.

- **Zero Argument:** For the logarithm to be defined, the argument \( \frac{x+2}{x^3} \) must be positive. However, we focus on where the logarithm might go to negative infinity, which occurs when the argument approaches zero. The equation \( \frac{x+2}{x^3} = 0 \) does not yield any real solutions, so there is no contribution to the asymptote from this condition.

### Step 3: Conclusion

The function \( g(x) \) has vertical asymptotes at the points where it is undefined:

- \( x = 1 \) (from the expression \( \frac{|x-1|}{x-1} \))
- \( x = 0 \) (from the logarithmic expression \( \ln\left(\frac{x+2}{x^3}\right) \))

Would you like more details or have any questions?

Here are 5 related questions you might consider:
1. How do you determine the domain of a function involving logarithms?
2. What are the conditions for a function to have a vertical asymptote?
3. How does the behavior of absolute value functions affect discontinuities?
4. Can a function have both a vertical asymptote and a hole at the same point?
5. What is the difference between a vertical asymptote and a removable discontinuity?

**Tip:** Always consider both components of a composite function when looking for vertical asymptotes: where the function is undefined and where the logarithmic part might approach negative infinity.

Discover how to determine the values of 𝑥 where the function 𝑔(𝑥) = ((|x-1|)/(x-1))*(ln((x+2)/x^3)) has vertical asymptotes. This detailed analysis covers absolute value functions, logarithmic expressions, and the conditions for vertical asymptotes in functions involving absolute values and logarithms. Suitable for advanced high school students and calculus enthusiasts.

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