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SOLVE THIS PROBLEM AND :Calculate the derivative, Show the tangent at a point, indicate if the discontinuity is 
Salvageable,Capture Screen with Area Algebraic, Graph, and Protocol of construction

### Solving the Limit

We are asked to find the limit:

\[
\lim_{{x \to 0}} 2x \cdot \ln(x^2)
\]

As explained earlier, this limit simplifies to:

\[
\lim_{{x \to 0}} 4x \cdot \ln(x) = 0
\]

So, the limit is **0**.

### Derivative Calculation

Let's calculate the derivative of the function \( f(x) = 2x \ln(x^2) \).

First, simplify the function using the logarithmic identity:

\[
f(x) = 4x \ln(x)
\]

Now, apply the product rule for differentiation:

\[
f'(x) = \frac{d}{dx} \left(4x \ln(x)\right) = 4 \cdot \frac{d}{dx} \left(x \ln(x)\right)
\]

Applying the product rule:

\[
f'(x) = 4 \left(\ln(x) + x \cdot \frac{1}{x}\right) = 4 \left(\ln(x) + 1\right)
\]

So, the derivative is:

\[
f'(x) = 4 \ln(x) + 4
\]

### Tangent at a Point

To find the equation of the tangent line at a point \( x = a \), we use the point-slope form of the equation of a line:

\[
y - f(a) = f'(a)(x - a)
\]

Let's choose \( x = 1 \) as a sample point:

1. Evaluate \( f(1) \):
   \[
   f(1) = 4 \times 1 \times \ln(1) = 4 \times 0 = 0
   \]
2. Evaluate \( f'(1) \):
   \[
   f'(1) = 4 \ln(1) + 4 = 4 \times 0 + 4 = 4
   \]

So the equation of the tangent line at \( x = 1 \) is:

\[
y - 0 = 4(x - 1)
\]

Simplifying:

\[
y = 4x - 4
\]

### Discontinuity

Since the function involves \( \ln(x) \), it is not defined for \( x \leq 0 \). However, on its domain \( (0, \infty) \), there is no discontinuity at \( x = 0 \) as the limit tends to 0 smoothly. Therefore, the discontinuity at \( x = 0 \) is not salvageable as the logarithmic function is not defined there.

### Graph and Algebraic Representation

To visually understand the function, a graph should be plotted for \( f(x) = 4x \ln(x) \) and its tangent line at \( x = 1 \).

Unfortunately, I cannot generate or capture screen shots directly, but you can graph it using a tool like GeoGebra or any graphing calculator by entering the function and the tangent equation:

1. **Graph**: \( f(x) = 4x \ln(x) \)
2. **Tangent line**: \( y = 4x - 4 \)
3. **Point of Tangency**: \( (1, 0) \)

### Summary of Steps

1. **Limit**: \( \lim_{{x \to 0}} 2x \ln(x^2) = 0 \).
2. **Derivative**: \( f'(x) = 4 \ln(x) + 4 \).
3. **Tangent Line**: \( y = 4x - 4 \) at \( x = 1 \).
4. **Discontinuity**: The discontinuity at \( x = 0 \) is not salvageable.

If you have further questions or need additional explanations, feel free to ask.

---

Here are five questions you might find useful:

1. What methods can be used to analyze the behavior of a function at points of discontinuity?
2. How does the product rule apply to logarithmic functions during differentiation?
3. What are the key characteristics of the natural logarithm function, especially near \( x = 0 \)?
4. Can you plot different tangent lines to the function at various points using graphing software?
5. What is the significance of a tangent line in understanding the behavior of a function near a given point?

**Tip:** Always verify if the function is defined over the entire domain before applying calculus operations.

Learn how to solve mathematical problems involving limits and derivatives. Calculate limits, find tangent lines, and analyze discontinuities with detailed step-by-step solutions. Suitable for advanced high school students.

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