Determine the nature of the function f(x)=(x^2(log(x+√(x^2+1))))/((x cos^2 x+tanx+x^7))

Name: Problem 161, Chapter 1: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-02-26T18:45:51-08:00
Duration: 1 min 4 s
Channel: Varsha Aggarwal
Description: Determine the nature of the function f(x)=(x^2(log(x+√(x^2+1))))/((x cos^2 x+tanx+x^7))

step 1 So, we have numerator equal, numerator equal even function into odd function and even into odd r

Determine the nature of the function...

Key Concepts

Inverse_Hyperbolic_Function Representation

The logarithmic term log(x + ?(x²+1)) can be recognized as the inverse hyperbolic sine function, as asinh(x) = log(x + ?(x²+1)). This transformation is valuable in analysis because it allows one to leverage known properties of inverse hyperbolic functions, such as their parity and asymptotic behavior.

Parity Analysis of Functions

Understanding whether a function is even, odd, or neither is a key concept in function analysis. In this context, by identifying that asinh(x) is an odd function and recognizing the even or odd nature of other algebraic factors (such as x² being even and terms like x or tan(x) being odd), one can determine the overall symmetry of the composite function. The multiplication of an even function with an odd function yields an odd function, and the ratio of two odd functions is even, which gives insight into the symmetry of the function.

Dominant Term Analysis in Rational Functions

When dealing with rational functions that combine polynomial and transcendental components, it is crucial to identify which terms dominate the behavior of the numerator and denominator, especially at extreme values of x. This involves comparing the degrees of polynomial expressions and the growth rates of functions like logarithmic or trigonometric functions to understand the function's growth or decay rates, which is essential for understanding limits and the overall behavior of the function.

Asymptotic Behavior

Analyzing the asymptotic behavior of a function involves studying its limits as x approaches infinity or zero. By determining how the dominant terms in the numerator and denominator behave, one can predict whether the function tends to a finite limit, grows without bound, or decays to zero. This type of analysis is important for understanding the global behavior of functions and for applications involving approximations and limits.

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