Final answer:
The limit of the sequence (2^n + 3^n)/(4^n - 5) as n tends to infinity is 0, focusing on the dominant terms 3^n in the numerator and 4^n in the denominator.
Step-by-step explanation:
To find the limit of the sequence (2^n + 3^n)/(4^n - 5) as n tends to infinity using the algebra of limits, let's focus on the dominant terms, which are those with the highest exponents. As n approaches infinity, the terms 2^n and 3^n become significantly smaller compared to 4^n, making the -5 negligible. The term 3^n grows faster than 2^n, so it will dominate the numerator. Similarly, 4^n will dominate the denominator. Thus, the limit can be approximated by comparing the dominant terms:
Limit of (2^n + 3^n)/(4^n - 5) as n tends to infinity ≈ Limit of 3^n/4^n as n tends to infinity
Using the properties of exponents, we can simplify this to:
Limit of (3/4)^n as n tends to infinity = 0
Therefore, the limit of the given sequence is 0.