Question: Evaluate the limit as n tends to infinity of [{log(1 + x)}^2 - {log(x)}^2].

Solution:
To find the limit of the given expression as n tends to infinity, we can simplify the expression and analyze its behavior.

Simplifying the expression:
Let's start by expanding the given expression:

[{log(1 + x)}^2 - {log(x)}^2]

Using the identity log(a) - log(b) = log(a/b), we can rewrite the expression as:

[{log((1 + x)/x)} * {log((1 + x)/x)}]

Simplifying further, we get:

[{log((1 + x)/(x))} * {log((1 + x)/(x))}]

Asymptotic Behavior:
To analyze the asymptotic behavior of the expression as n tends to infinity, we can consider the behavior of the individual logarithmic terms.

Logarithmic term 1: log((1 + x)/(x))
As x approaches infinity, the term (1 + x)/x approaches 1, since the x term dominates the 1 term. Therefore, we have:

log((1 + x)/(x)) ≈ log(1) = 0

Logarithmic term 2: log((1 + x)/(x))
Similarly, as x approaches infinity, the term (1 + x)/x approaches 1, and we have:

log((1 + x)/(x)) ≈ log(1) = 0

Final expression:
Substituting the asymptotic behavior of the logarithmic terms back into the original expression, we get:

[{log((1 + x)/(x))} * {log((1 + x)/(x))}] ≈ [0 * 0] = 0

Conclusion:
Therefore, as n tends to infinity, the given expression [{log(1 + x)}^2 - {log(x)}^2] approaches 0.