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**Question 1.1.2**  
- *Find the limit of the following function as \( x \) approaches infinity:*
\[
f(x) = \frac{3x^2 + 2x - 1}{5x^2 - x + 4}
\]

**Solution:**
- The degrees of the polynomials in the numerator and denominator are the same (degree 2).
- When the degrees are the same, the limit as \( x \) approaches infinity is the ratio of the leading coefficients.

\[
\lim_{x \to \infty} \frac{3x^2 + 2x - 1}{5x^2 - x + 4} = \frac{3}{5}
\]

So, the limit is \(\frac{3}{5}\).

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**Question 1.2**  
- *Differentiate the following function with respect to \( x \):*
\[
g(x) = e^{2x} \cdot \ln(x^2 + 1)
\]

**Solution:**
- Apply the product rule of differentiation: \((u \cdot v)' = u'v + uv'\).
  
Let \( u(x) = e^{2x} \) and \( v(x) = \ln(x^2 + 1) \).

1. Differentiate \( u(x) \):
\[
u'(x) = \frac{d}{dx} e^{2x} = 2e^{2x}
\]

2. Differentiate \( v(x) \):
\[
v'(x) = \frac{d}{dx} \ln(x^2 + 1) = \frac{1}{x^2 + 1} \cdot \frac{d}{dx}(x^2 + 1) = \frac{2x}{x^2 + 1}
\]

3. Apply the product rule:
\[
g'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
\[
g'(x) = 2e^{2x} \cdot \ln(x^2 + 1) + e^{2x} \cdot \frac{2x}{x^2 + 1}
\]

Thus, the derivative is:
\[
g'(x) = 2e^{2x} \ln(x^2 + 1) + \frac{2xe^{2x}}{x^2 + 1}
\]

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Do you have any questions or would you like further details on any of these solutions? 

Here are some related questions you might find useful:

1. What is the limit of a function when the degrees of the numerator and denominator are different?
2. How do you differentiate a function that is a product of three functions?
3. Can you apply L'Hôpital's Rule to the limit problem in 1.1.2?
4. What is the integral of the function in question 1.2?
5. How would you solve the limit if the coefficients were not constants but functions of \(x\)?

**Tip:** When dealing with limits involving polynomials, always compare the degrees of the numerator and denominator first. This often simplifies the process.

Explore detailed solutions to mathematical questions 1.2 and 1.1.2 involving limits and differentiation. Learn how to find limits of polynomial functions and differentiate product functions step-by-step. Suitable for advanced high school students and beyond.

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