Final Answer:
The limit of n (√(1 + 1/n) - 1) as n approaches infinity is 1/2.
Step-by-step explanation:
To find the limit, we can use algebraic manipulation and the concept of limits. Begin by multiplying both the numerator and denominator by the conjugate of the expression, which in this case is (√(1 + 1/n) + 1). This step helps eliminate the square root in the numerator. After the manipulation, you'll end up with an expression that can be simplified further.
n * (√(1 + 1/n) - 1) / (√(1 + 1/n) + 1)
Simplifying the expression inside the limit by canceling out terms yields:
1 / (2√(1 + 1/n))
Further simplification results in:
1/2
Thus, the limit of the given expression as n approaches infinity is 1/2. This process involves careful algebraic manipulation and application of limit rules to simplify the expression and find the ultimate limit.