The value of limx →∞[(1^1 / x+2^1 / x+3^1 / x+…+n^1 / x)/(n)]^n x is (A) n ! (B) n (C) (n-1) ! (

Step 1: Given the limit $lim _{x rightarrow infty}left[frac{1^{1 / x}+2^{1 / x}+3^{1 / x}+ldots+

The value of limx →∞[(1^1 / x+2^1 / x+3^1 / x+…+n^1 /...

Key Concepts

Exponential Limit Theorem

This theorem discusses the behavior of expressions in the form (1 + a/n)^n as n becomes large, where such an expression converges to the exponential function e^a. It is a fundamental concept in calculus, enabling the evaluation of limits by transforming compound expressions into simpler exponential forms.

Series Expansion for Small Arguments

When a variable in the exponent is very small, functions such as a^(1/x) can be approximated using their first-order Taylor expansion. For large x, a^(1/x) is approximated by 1 + (log a)/x, which simplifies complex expressions and allows for easier limit evaluation.

Properties of Logarithms

Logarithms convert multiplicative relationships into additive ones, making them a powerful tool in limit evaluation. By summing logarithms, one can transform the product of terms into a sum that is easier to approximate, and subsequently exponentiating the sum recovers the product, often revealing factorial forms in limit problems.

Factorial Function

The factorial function, defined as the product of all positive integers up to a given number, frequently appears in contexts where compounded products are involved. In limit problems, especially those involving logarithmic summations, the result can often be expressed as a factorial, highlighting the connection between continuous approximations and discrete mathematics.

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