limx →1 (tan(x-1) ·loge x^x-1)/(|x-1|^3) is equal to (A) 1 (B) -1 (C) 3 (D) None of these

Name: Problem 27, Chapter 1: Differential Calculus for JEE Main and Advanced
Uploaded: 2022-01-30T14:20:33-08:00
Duration: 4 min 45 s
Channel: Harmender Singh Yadav
Description: limx →1 (tan(x-1) ·loge x^x-1)/(|x-1|^3) is equal to (A) 1 (B) -1 (C) 3 (D) None of these

step 1 In this question, we need to evaluate the limit, that is, limit X turns to 1, tangent x minus 1

Limx →1 (tan(x-1) ·loge x^x-1)/(|x-1|^3) is equal to (A) 1...

Key Concepts

Effect of Absolute Value in Limit Evaluations

The presence of an absolute value in the limit expression, particularly in the denominator, alters the behavior of the function depending on whether the variable approaches the limit from above or below. Absolute values remove the sign, which can lead to discrepancies when the numerator retains sign information. Consequently, when evaluating such limits, it is important to check the limit from both sides to determine if the overall limit exists or if the left- and right-hand limits differ.

Limit of Functions and One-Sided Limits

This concept examines how a function behaves as its variable approaches a point from either side. In the context of limits, it is important to consider that approaching from the left or right (i.e. negative or positive increments) may yield different values, especially when the function or its components are sensitive to the sign of the input. This is particularly relevant when absolute values are involved, as they can mask the sign in certain parts of the expression, potentially leading to different behaviors from the two sides.

Taylor Series Expansion

Taylor series provide approximations of functions around a specific point by using the function's derivatives at that point. In limit problems, particularly when dealing with functions like tangent and logarithm near zero, the first few terms of their Taylor expansions can be used to simplify complicated expressions. This method transforms the problem into one involving polynomials, making it easier to compare the degrees of the numerator and denominator and to identify the dominant term.

Transcript

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