The value of limx →1 (x^n+x^n-1+x^n-2+…+x^2+x-n)/(x-1) is (A) (n(n+1))/(2) (B) 0 (C) 1 (D) n

Name: Problem 33, Chapter 11: A Complete Resource Book in Mathematics for JEE Main
Uploaded: 2021-10-03T17:17:29-08:00
Duration: 2 min 20 s
Channel: Abhijith V
Description: The value of limx →1 (x^n+x^n-1+x^n-2+…+x^2+x-n)/(x-1) is (A) (n(n+1))/(2) (B) 0 (C) 1 (D) n

step 1 So here we need to find the value of the limit, limit extends to 1, x to the power n plus x to t

The value of limx →1 (x^n+x^n-1+x^n-2+…+x^2+x-n)/(x-1) is...

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Key Concepts

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence, where each term increases by a fixed difference. Recognizing the structure of an arithmetic series is important in deriving closed-form expressions, such as the sum of consecutive integers, which often appears in problems involving sums and limits.

Indeterminate Forms

Indeterminate forms occur when substituting a value into an expression results in an undefined expression, such as 0/0. Recognizing indeterminate forms signals that conventional substitution is not sufficient, and that techniques like algebraic manipulation or calculus methods must be employed.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that initially appear as indeterminate forms, such as 0/0 or ?/?. The rule states that under appropriate conditions, the limit of a ratio of functions can be found by differentiating the numerator and the denominator separately and then taking the limit of the resulting quotient.

Differentiation of Polynomials

Differentiation of polynomials involves applying the power rule to find the derivative of each term in the polynomial. This process is useful in limit evaluation, especially when using L'Hôpital's Rule, because it simplifies the expression and makes it possible to compute the limit directly.

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