Evaluate this limit:

limit {(2+√2)n}
n→∞
where, {x} = x - floor(x)

The limit you provided can be evaluated using the following steps:

Step 1: Identify the dominant term

In the expression $(2+\sqrt{2}{)}^n$, the dominant term as n approaches infinity is the term with the highest power of n. In this case, all terms have the power of n, so the dominant term is the entire expression itself.

Step 2: Simplify the expression

Since the value of a constant factor multiplied by a dominant term doesn't affect the limit, we can ignore the constant factor 2. We are left with:

limit {(√2)^n} n→∞

Step 3: Apply the properties of fractional exponents

(√2)^n can be written as 2^(n/2). As n approaches infinity, n/2 also approaches infinity. Therefore:

limit (2^(n/2)) n→∞

Step 4: Evaluate the limit

Any positive constant raised to the power of infinity will result in infinity. Therefore:

limit (2^(n/2)) n→∞ = ∞

Conclusion:

The limit of {(2+√2)^n} as n approaches infinity, where {x} = x - floor(x), is infinity.

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