 ## Book: RD Sharma - Mathematics (Volume 1)

### Chapter: 9. Continuity

#### Subject: Maths - Class 12th

##### Q. No. 3 of Exercise 9.2

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##### Find the points of discontinuity, if any, of the following functions : Basic Idea:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if : where h is a very small ‘+ve’ no.

i.e. left hand limit as x c (LHL) = right hand limit as x c (RHL) = value of function at x = c.

This is very precise, using our fundamental idea of limit from class 11 we can summarise it as, A function is continuous at x = c if : Here we have, …Equation 1

Note: [for changing the expression used identity:– (a2–b2) = (a+b)(a–b)]

Note: x – 2 is cancelled from numerator and denominator only because x ≠ 2, else we can’t cancel them

The function is defined for all real numbers, so we need to comment about its continuity for all numbers in its domain ( domain = set of numbers for which f is defined )

Function is changing its nature (or expression) at x = 2, So we need to check its continuity at x = 2 first.

Clearly,

f(2) = 16 [using eqn 1] Note: (x – 2) is cancelled as x ≠ 2 but x 2

Clearly, f(x) is continuous at x = 2.

Let c be any real number such that c ≠ 0

f(c) = [using eqn 1] Clearly, f(x) is continuous for all real x

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