To evaluate the limit:

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

Step-by-step solution:

1. Identify the dominant terms:

   • In the numerator $3n^2 + 4n + 7$, the dominant term as $n$ approaches infinity is $3n^2$, since it has the highest degree.
   • In the denominator $2 + 3n + 4n^2$, the dominant term is $4n^2$.

2. Simplify the fraction by dividing the numerator and denominator by $n^2$ (the highest power of $n$ present in the expression):

   $\frac{3n^2 + 4n + 7}{2 + 3n + 4n^2} = \frac{3n^2/n^2 + 4n/n^2 + 7/n^2}{2/n^2 + 3n/n^2 + 4n^2/n^2}$

   This simplifies to:

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

3. Take the limit as $n \to \infty$:

   • As $n \to \infty$, $\frac{4}{n} \to 0$, $\frac{7}{n^2} \to 0$, $\frac{2}{n^2} \to 0$, and $\frac{3}{n} \to 0$.
   • This leaves us with:

   $\lim_{n \to \infty} \frac{3 + 0 + 0}{0 + 0 + 4} = \frac{3}{4}$

Final answer:

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

Would you like further explanation on any step, or have any questions about similar limits?

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Here are 5 related questions:

1. How do you evaluate limits of rational functions involving higher powers of $n$?
2. What is the limit of the expression $\frac{n^2 + 2n}{n^2 + 3}$ as $n \to \infty$?
3. How do you approach limits involving exponential and polynomial terms?
4. What happens if the degrees of the numerator and denominator are equal in a rational function?
5. Can you explain the intuition behind dividing both the numerator and denominator by the highest power of $n$?

Tip: When evaluating limits for rational functions, always identify the terms with the highest powers of $n$, and divide through by those terms to simplify the limit evaluation.