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Tamil Nadu Board of Secondary EducationHSC Science Class 11

Examine the continuity of the following: x . log x - Mathematics

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Question

Examine the continuity of the following:

x . log x

Sum
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Solution

Let f(x) = x log x

The function f(x) is defined in the open interval `(0, oo)` since log x is defined for x > 0.

Let x0 be an arbitrary point in `(0, oo)`.

Then `lim_(x -> x_0) f(x) =  lim_(x -> x_0)  x log x`

= x0 log x0

f(x0) = x0 log x0

From equation (1) and (2) we have

`lim_(x -> x_0) f(x) = f(x_0)`

∴ The limit of the function f(x) exists at x = x0 and is equal to the value of the function.

Since x0 is an arbitrary point the above is true for all points in `(0, oo)`.

∴ f(x) is continuous at all points of `(0, oo)`.

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Chapter 9: Differential Calculus - Limits and Continuity - Exercise 9.5 [Page 127]

APPEARS IN

Samacheer Kalvi Mathematics - Volume 1 and 2 [English] Class 11 TN Board
Chapter 9 Differential Calculus - Limits and Continuity
Exercise 9.5 | Q 2. (v) | Page 127

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