Final answer:
The limit of (1 + 3/x)^2 as x approaches negative infinity is 1, as the term 3/x approaches 0 making the expression inside the parentheses approach 1.
Step-by-step explanation:
To find the limit of (1 + 3/x)^2 as x approaches negative infinity, you would recognize that as x gets larger in magnitude but negative, the term 3/x approaches 0. Therefore, the expression inside the parentheses approaches 1.
Since any number to the zeroth power equals 1, this leads us to understand the limit will approach 1. Specifically, here the function behaves similarly to an asymptote, demonstrating that as x approaches negative infinity, the function approaches a limit value which, in this case, is 1.
This limit simplifies the function's behavior at extreme values of x, providing a clearer understanding of its end behavior, which is an important concept in the study of limits and calculus.
To find the limit of (1 + 3/x)²ˣ as x approaches negative infinity, we can rewrite the expression as ((1 + 3/x)^(x/(3/x)))^3. We can then rewrite (1 + 3/x)^(x/(3/x)) as e^(3).
As x approaches negative infinity, the expression e^(3) remains constant. Therefore, the limit is e^(3).