Final answer:
By reformulating the limit expression and comparing it with a known limit identity, it's found that the values of a and b that satisfy the given limit equation are 1 and 0, respectively.
Step-by-step explanation:
To find the values of a and b that make the equation limx→[infinity] (1+ a/x + b/ x² ) ²ˆ = e² true, we need to manipulate the equation in a way that reveals a clear limit as x goes to infinity and apply known limits such as the fact that limx→[infinity] (1 + 1/x)ˆˆ = e.
Rewriting the given limit expression to fit into the above form, we have:
limx→[infinity] ((1 + a/x + b/ x²)²ˆ) = limx→[infinity] ((1 + 1/(x/a) + (b/a²)/(x/a)²)²ˆˆ)
Comparing this equation to the identity, we can see that if we choose a such that x/a becomes 1/x, and b such that it cancels out, we get the form of the known limit. We set a = 1 and b = 0, giving us:
limx→[infinity] (1 + 1/x + 0/ x² ) ²ˆ which simplifies to the known limit and is equal to e².
Therefore, the values of a and b are 1 and 0, respectively.