Locate the discontinuities of the function. (Use n as a arbitrary integer.) f(x) = ln(tan^2(x))

To find the discontinuities of the function \( f(x) = \ln(\tan^2(x)) \), we need to examine where the argument of the logarithm, \( \tan^2(x) \), is undefined or non-positive. Let's break this down step by step:

### 1. **Argument of the Logarithm Must Be Positive**:
Since \( f(x) = \ln(\tan^2(x)) \), the function is only defined when \( \tan^2(x) > 0 \). This is because the logarithm function, \( \ln(x) \), is only defined for \( x > 0 \).

- \( \tan^2(x) \) is positive for all \( x \) where \( \tan(x) \) is defined and nonzero.

### 2. **Tangent Function Discontinuities**:
The tangent function, \( \tan(x) \), is undefined where \( \cos(x) = 0 \). This occurs at:
\[
x = \frac{\pi}{2} + n\pi \quad \text{for} \quad n \in \mathbb{Z}
\]
Thus, \( f(x) \) will be undefined at these points because \( \tan(x) \) approaches infinity or negative infinity.

### 3. **Where \( \tan^2(x) = 0 \)**:
For \( f(x) = \ln(\tan^2(x)) \), the argument \( \tan^2(x) \) must also not be zero, since \( \ln(0) \) is undefined. \( \tan^2(x) = 0 \) when \( \tan(x) = 0 \), which occurs at:
\[
x = n\pi \quad \text{for} \quad n \in \mathbb{Z}
\]
Thus, \( f(x) \) will also be undefined at these points because \( \tan(x) = 0 \) makes the argument of the logarithm zero.

### 4. **Summary of Discontinuities**:
The function \( f(x) = \ln(\tan^2(x)) \) is discontinuous at the points where:
\[
x = n\pi \quad \text{and} \quad x = \frac{\pi}{2} + n\pi \quad \text{for} \quad n \in \mathbb{Z}
\]
In other words, the discontinuities occur at all multiples of \( \pi \) and at odd multiples of \( \frac{\pi}{2} \).

Would you like more details on this, or do you have any questions?

### Related Questions:
1. What is the domain of the function \( f(x) = \ln(\tan^2(x)) \)?
2. How do you find the asymptotes of \( \tan(x) \)?
3. What is the behavior of \( \tan(x) \) near its discontinuities?
4. How do you calculate limits involving logarithmic functions?
5. How do you handle functions with discontinuities in calculus?

### Tip:
Always check for points where trigonometric functions are undefined or equal to zero when working with logarithmic expressions involving them.

Learn how to locate the discontinuities of the function f(x) = ln(tan^2(x)). This problem explains the conditions where the tangent function is undefined or zero and how it leads to discontinuities in the logarithmic function. Suitable for advanced high school students in Grades 11-12.

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