The given problem involves determining the limiting value of the sequence defined as:

$T_n = 2 + \frac{7}{an + b}, \quad a \neq 0$

Steps to Determine the Limiting Value:

1. As $n \to \infty$, the denominator $an + b$ grows indefinitely because $a \neq 0$.
2. Consequently, the term $\frac{7}{an + b} \to 0$ because the numerator (7) remains constant while the denominator increases without bound.
3. Thus, the sequence simplifies as $n \to \infty$: $T_n = 2 + \frac{7}{an + b} \to 2 + 0 = 2$

Conclusion:

The limiting value of the sequence is:

$\boxed{2}$

Would you like additional clarification or a related example? Here are some related questions for deeper understanding:

1. What happens if $a = 0$ in this sequence?
2. How can we verify the limit of the sequence graphically?
3. What if the numerator changes to a variable instead of a constant?
4. Can this sequence converge to a value other than 2 for different forms of the denominator?
5. How does the value of $b$ affect the speed of convergence?

Tip: When evaluating the limiting behavior of sequences, simplify expressions step by step and focus on how terms behave as $n$ approaches infinity.