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This document contains three mathematical problems. Below is their breakdown along with solutions:

---

### **Problem 1**: Find the limit  
\[
\lim_{x \to \infty} \frac{53x^3 - 27x + 95}{48x^3 - 47}
\]

### **Solution**:
To solve this, divide the numerator and denominator by \(x^3\) (the highest power of \(x\)):

\[
\frac{53x^3 - 27x + 95}{48x^3 - 47} = \frac{53 - \frac{27}{x^2} + \frac{95}{x^3}}{48 - \frac{47}{x^3}}
\]

As \(x \to \infty\), terms involving \(1/x\) approach 0. Thus:

\[
\lim_{x \to \infty} \frac{53x^3 - 27x + 95}{48x^3 - 47} = \frac{53}{48}
\]

**Answer**: \(\frac{53}{48}\)

---

### **Problem 2**: Solve the differential equation  
\[
-53y(x) - 48x + y'(x) + 28 = 0
\]

### **Solution**:
This is a first-order linear differential equation. Rewrite it as:

\[
y'(x) = 53y(x) + 48x - 28
\]

#### Step 1: Solve the homogeneous equation
\[
y'(x) - 53y(x) = 0
\]

This has the solution \(y_h(x) = Ce^{53x}\), where \(C\) is an arbitrary constant.

#### Step 2: Find a particular solution
We use the method of undetermined coefficients. Assume \(y_p(x) = Ax + B\). Substituting into the equation:

\[
y_p'(x) = A \quad \text{and} \quad y_p(x) = Ax + B
\]

Substituting \(y_p(x)\) and \(y_p'(x)\) into the original equation:

\[
A - 53(Ax + B) = -48x + 28
\]

Equating coefficients of \(x\) and constants:

1. Coefficient of \(x\): \(-53A = -48 \implies A = \frac{48}{53}\)
2. Constant: \(A - 53B = 28 \implies \frac{48}{53} - 53B = 28 \implies B = \frac{48}{53 \cdot 53} - \frac{28}{53}\)

Thus, the general solution is:

\[
y(x) = Ce^{53x} + \frac{48}{53}x + \text{(simplify \(B\)) for constant.}
\]

---

### **Problem 3**: Plot a polar graph  

#### Given polar function:
\[
r(t) = \frac{100 (2 - \sin(7t) - \frac{1}{2}\cos(30t))}{100 + (t - \frac{\pi}{2})^8}, \quad t \in \left[-\frac{\pi}{2}, \frac{3\pi}{2}\right]
\]

#### Instructions:
1. Use `PolarPlot` in a mathematical plotting software (e.g., Mathematica).
2. Customize the plot using:
   - `Axes -> False`
   - `PlotStyle -> {Thickness[.007], RGBColor[0, 1, 0]}`

---

Would you like a detailed explanation of any step?  

### Further Questions:
1. How can the general solution of the differential equation be verified?
2. What does dividing terms by \(x^3\) achieve in Problem 1?
3. Why does the method of undetermined coefficients work for Problem 2?
4. How does modifying the plot style in `PolarPlot` improve visualization?
5. How would the solution in Problem 1 change if the limit was \(x \to -\infty\)?

---

### Tip:
When solving limits, always simplify by dividing by the highest power of the variable in the denominator to isolate the dominant terms.

Find the limit lim_{x \to \infty} \frac{53x^3 - 27x + 95}{48x^3 - 47}

Learn how to find the limit of the rational function \(\frac{53x^3 - 27x + 95}{48x^3 - 47}\) as x approaches infinity. This problem covers key concepts like polynomial limits and simplifying expressions at infinity. Suitable for students in Grades 11-12.

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