Find the value of limx →0((1-cosx cos2 x cos3 x)/(x^2)) |...

Name: Problem 7, Chapter 5: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-03-28T06:37:46-08:00
Duration: 3 min 19 s
Channel: Heena Mahenoor
Description: Find the value of limx →0((1-cosx cos2 x cos3 x)/(x^2))

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Key Concepts

Limits

The concept of limits involves studying the behavior of a function as the input approaches a particular value. In calculus, limits are essential for understanding function continuity and the foundation of derivatives and integrals. Evaluating limits, especially as a variable tends towards a specific value, allows for determining the behavior of functions near points where immediate evaluation might be indeterminate.

Taylor Series Expansion

Taylor series expansion is a method to approximate functions near a given point using an infinite sum of derivative terms evaluated at that point. For trigonometric functions like cosine, the Taylor series provides a polynomial approximation (cos x ? 1 - x²/2 for small x). This approximation is crucial in limit evaluation because it simplifies the expression by focusing on the leading order behavior.

Small-Angle Approximations

Small-angle approximations are techniques used when dealing with trigonometric functions for angles close to zero. By approximating functions such as cosine with their Taylor series expansions, one can simplify the product or sum of trigonometric functions to obtain a more manageable, often polynomial, form. This approximation underpins the process of evaluating limits involving trigonometric functions at small angles.

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