Let f ( x ) = lim 0 cos ( x ) - x ( 2 ) sin ( x - 1 ) 1 + x ( 2) ( x - 1 ) , x . Consider the following two statements: I. f(x) is discontinuous at x = 1 II. f(x) is continuous at x = -1 Then, [2026]

Question

Let $f (x) = \lim_{θ \to 0} (\frac{\cos (π x) - x^{(\frac{2}{θ})} \sin (x - 1)}{1 + x^{(\frac{2}{θ})} (x - 1)}), x \in ℝ .$

Consider the following two statements:

I. f(x) is discontinuous at x = 1

II. f(x) is continuous at x = -1

Then, [2026]

Smart Achievers · Accepted Answer

(3)

$f (x)$ $=$ ${\begin{matrix} \cos (π x) & x \to 1^{-} \\ \frac{- \sin (x - 1)}{x - 1} & x \to 1^{+} \end{matrix}$

$RHL = \lim_{x \to 1} \frac{- \sin (x - 1)}{x - 1} = - 1$

$LHL = \lim_{x \to 1} \cos (π x) = - 1, f (1) = - 1$

$f(x) is continuous at x = 1$

$f (x)$ $=$ ${\begin{matrix} \frac{- \sin (x - 1)}{- (x - 1)} & x \to - 1^{-} \\ \cos (π x) & x \to - 1^{+} \end{matrix}$

$RHL = \lim_{x \to - 1} \cos (π x) = - 1$

$LHL = \lim_{x \to - 1} \frac{- \sin (x - 1)}{x - 1} = \frac{\sin 2}{- 2}$

$f(x) is discontinuous at x = - 1$

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