Understanding the Limit of a Polynomial Expression as x Approaches 1

When dealing with polynomials and their limits, one often encounters expressions of the form:

[ lim_{x to 1} frac{x^n x^{n-1} x^{n-2} cdots x - n}{x - 1} ]

This particular problem is a fundamental concept in mathematics, especially relevant for high school and undergraduate students. We will break it down to simplify and find the value of the given limit without delving into complex calculus or advanced mathematics.

Breaking Down the Expression

Let's denote the limit as ( L_n ). We begin with:

[ L_n lim_{x to 1} frac{1}{x - 1} sum_{k1}^{n} x^k - n ]

Let's simplify the expression inside the summation first:

[ L_n lim_{x to 1} sum_{k1}^{n} frac{x^k - 1}{x - 1} ]

We know from the geometric series formula that:

[ frac{x^k - 1}{x - 1} sum_{t0}^{k-1} x^t ]

Therefore, we can rewrite the limit as:

[ L_n lim_{x to 1} sum_{k1}^{n} sum_{t0}^{k-1} x^t ]

This expression can be simplified further by recognizing the double summation as a single summation. Each term ( x^t ) appears in ( k - t ) terms for ( t 0, 1, 2, ldots, k-1 ). Summing over ( k ) and ( t ), we get:

[ L_n sum_{k1}^{n} k ]

This simplifies to:

[ L_n frac{1}{2} n (n 1) ]

Alternative Approaches

Alternatively, one can use the concept of L'H?pital's Rule when the numerator and denominator both approach 0 as ( x ) approaches 1. This will yield the same result:

[ L lim_{x to 1} frac{sum_{k1}^{n} x^k - n}{x - 1} ]

Differentiating both the numerator and denominator with respect to ( x ) gives:

[ L lim_{x to 1} frac{sum_{k1}^{n} k x^{k-1}}{1} sum_{k1}^{n} k frac{n(n 1)}{2} ]

This confirms the result using a different method.

Synthetic Division and Evaluation

Another method to solve the problem is through synthetic division. The synthetic division of the expression ( x^n x^{n-1} x^{n-2} cdots x - n ) by ( x - 1 ) yields:

[ x^{n-1} 2x^{n-2} 3x^{n-3} cdots n ]

Evaluating this expression as ( x ) approaches 1, we get:

[ 1 2 3 cdots n frac{n(n 1)}{2} ]

This is equivalent to the result obtained using L'H?pital's Rule.

Conclusion

The value of the limit ( lim_{x to 1} frac{x^n x^{n-1} x^{n-2} cdots x - n}{x - 1} ) is ( frac{n(n 1)}{2} ). This expression is often encountered in mathematical problems and can be solved using various methods, including the geometric series, L'H?pital's Rule, and synthetic division.

The expression ( frac{x^n - 1}{x - 1} ) simplifies to ( n ) as ( x ) approaches 1, making the given limit straightforward to evaluate.