To evaluate the limit

$\lim_{n \to \infty} \frac{3n^2 + 4n + 7}{2 + 3n + 4n^2}$

we need to examine the behavior of the function as $n$ approaches infinity.

Step-by-Step Solution:

1. Identify the highest powers of $n$:

   • In the numerator, the highest power of $n$ is $n^2$ (from $3n^2$).
   • In the denominator, the highest power of $n$ is also $n^2$ (from $4n^2$).

2. Factor out the highest power of $n$ from both the numerator and the denominator:

   • Numerator: Factor out $n^2$ from the terms $3n^2 + 4n + 7$: $3n^2 + 4n + 7 = n^2 \left(3 + \frac{4}{n} + \frac{7}{n^2}\right)$
   • Denominator: Factor out $n^2$ from the terms $2 + 3n + 4n^2$: $2 + 3n + 4n^2 = n^2 \left(\frac{2}{n^2} + \frac{3}{n} + 4\right)$

3. Simplify the expression: Now, substitute these factored forms back into the original expression: $\frac{n^2 \left(3 + \frac{4}{n} + \frac{7}{n^2}\right)}{n^2 \left(\frac{2}{n^2} + \frac{3}{n} + 4\right)}$ The $n^2$ terms cancel out, leaving: $\frac{3 + \frac{4}{n} + \frac{7}{n^2}}{\frac{2}{n^2} + \frac{3}{n} + 4}$

4. Evaluate the limit as $n \to \infty$:

   • As $n \to \infty$, the terms $\frac{4}{n}$, $\frac{7}{n^2}$, $\frac{2}{n^2}$, and $\frac{3}{n}$ all approach zero.
   • This simplifies the expression to: $\frac{3 + 0 + 0}{0 + 0 + 4} = \frac{3}{4}$

Final Answer:

$\lim_{n \to \infty} \frac{3n^2 + 4n + 7}{2 + 3n + 4n^2} = \frac{3}{4}$

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Would you like more details or have any questions about this process?

Further Exploration:

1. How would the result change if the powers of $n$ in the numerator or denominator were different?
2. What happens if we have a higher degree polynomial in the denominator compared to the numerator?
3. How would this problem look if we had a constant in both the numerator and denominator (instead of polynomials)?
4. How does this concept apply when dealing with rational functions in general?
5. Could you use L'Hopital's Rule to solve this type of limit?

Tip: When evaluating limits of rational functions with polynomials, always look for the highest power of $n$ in both the numerator and denominator and simplify accordingly.