First slide

Continuity

Question

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

Moderate

A

-1
B

1
C

0
D

2

Solution

LHL $\lim_{x \to 0} β + [\frac{\sin x - x}{x^{3}}] \Rightarrow u s e l h o s p i t a l r u l e = β + \lim_{x \to 0} [\frac{\cos x - 1}{3 x^{2}}] = β + \lim_{x \to 0} [\frac{- \sin x}{6 x}] = β - 1$
RHL= $α + \frac{0}{x} = α$
$\begin{array}{l} α + 0 = 2 = β - 1 \\ α = β - 1 \Rightarrow β - α = 1 \end{array}$

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