First slide

Continuity

Question

$Let f : R \to R be defined by f (x) = \{\begin{array}{cc} α + \frac{\sin [x]}{x} & if x > 0 \\ 2 & if x = 0 Where [x] denotes the \\ β + [\frac{\sin x - x}{x^{3}}] x < 0 \end{array}$ $integral part of x . If f is continuous at x = 0 then β - α =$

Moderate

A

-1
B

1
C

0
D

2

Solution

$\underset{x \to 0}{L t} f (x) = f (0)$
$α + 0 = 2 = β - 1$
$\Rightarrow α = β - 1 \Rightarrow β - α = 1$

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