f(x)= (sin{x})/({x}), {x} ≠0 k {x}=0 f(x) can never be continuous for any value of K.

Name: Problem 5, Chapter 2: Differential Calculus for JEE Main and Advanced
Uploaded: 2021-11-23T05:41:55-08:00
Duration: 48 s
Channel: Hast Aggarwal
Description: f(x)= (sin{x})/({x}), {x} ≠0 k {x}=0 f(x) can never be continuous for any value of K.

step 1 So in this question we have been asked f of x is equal to is equal to sine of kind of x divided

F(x)= (sin{x})/({x}), {x} ≠0 k {x}=0 f(x) can...

$\mathrm{f}(\mathrm{x})= \begin{cases}\frac{\sin \{\mathrm{x}\}}{\{\mathrm{x}\}}, & \{\mathrm{x}\} \neq 0 \\ \mathrm{k} & \{\mathrm{x}\}=0\end{cases}$
$\mathrm{f}(\mathrm{x})$ can never be continuous for any value of $\mathrm{K}$.

Key Concepts

Trigonometric Limits

Trigonometric limits, such as the well-known limit of sin(x)/x as x approaches zero, provide fundamental tools in calculus for handling functions that involve trigonometric expressions. These limits are essential when determining the behavior of functions near points of potential discontinuity.

Fractional Part Function

The fractional part function, which yields the non-integer part of a real number, introduces periodic behavior and can create complications in limit analysis, particularly at points where the function resets. Understanding its properties is essential for analyzing functions that depend on the fractional part.

Piecewise Functions

Piecewise functions are defined by different expressions over different parts of their domain. Analyzing such functions involves studying the behavior of each piece, especially at the boundaries between the pieces, to assess properties like continuity or differentiability.

Limits

The concept of limits examines the behavior of a function as its argument approaches a particular value. In the context of piecewise functions, limits are crucial for determining if a potential discontinuity can be removed by correctly defining the function at that point.

Continuity

Continuity of a function requires that the function's value at a point equals the limit of the function as the input approaches that point. In analysis, verifying continuity means checking for equality between the function's defined value at the point and the limit taken from any path leading to that point.

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