If the function \(\rm f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {a + bx,\;\;}&{x < 1}\\ {5,}&{x = 1}\\ {b - ax,}&{x > 1} \end{array}} \right.\) is continuous, then what is the value of (a + b)?

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NDA (Held On: 18 Apr 2021) Maths Previous Year paper

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Answer 1

Concept:

For the function to be continuous:

LHL = RHL = f(x)

Where LHL = \(\rm\mathop {\lim }\limits_{α \to 0}\)f(x - α) and RHL = \(\rm\mathop {\lim }\limits_{α \to 0}\)f(x + α)

Calculation:

Given that f(x) is continuous function

LHL = f(x) = RHL

\(\rm\mathop {\lim }\limits_{α \to 0}\) f(1 - α) = f(1)

\(\rm\mathop {\lim }\limits_{α \to 0}\) [a + b(1 - α)] = 5

\(\rm\mathop {\lim }\limits_{α \to 0}\) [a + b - bα] = 5

a + b = 5

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