Let -x- denotes the greatest integer less than or equal to x- Then- the value of - for which the function fx-sin-x2-x2- x- 0 - x-0 is continuous at x - 0- is

The correct option is C α = sin (1) We have, #160; f(x) = #160;sin[ x]^2[ x]^2 α #160;, x≠ 0 Now, #160; x→ 0limsin[ x]^2[ x^2]=sin( 1) ...

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Standard XII

Mathematics

Domain

Let [x] den...

Question

Let $[x]$ denotes the greatest integer less than or equal to $x$ . Then, the value of $α$ for which the function $f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} \frac{sin [- x^{2}]}{[- x^{2}]} & x \neq 0 α & x = 0 \end{matrix}$ is continuous at $x = 0$ , is

α = 0

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α = sin (- 1)

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α = sin (1)

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α = 1

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Solution

The correct option is C $α = sin (1)$
We have,
$f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} \frac{sin [- x]^{2}}{[- x]^{2}} α \end{matrix}, x \neq 0$

Now,
$lim x \to 0 \frac{sin [- x]^{2}}{[- x^{2}]} = \frac{sin (- 1)}{(- 1)} = sin (1)$

Since, $f (x)$ is continuous at $x = 0$

$\Rightarrow lim x \to 0 f (x) = f (0) \Rightarrow sin (1) = α$

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Let [x] denotes the greatest integer less than or equal to x. Then, the value of α for which the function f(x)=⎧⎪⎨⎪⎩sin[−x2][−x2]x≠0αx=0 is continuous at x=0, is

The correct option is C α=sin(1)We have, f(x)=⎧⎪⎨⎪⎩sin[−x]2[−x]2α,x≠0Now, limx→0sin[−x]2[−x2]=sin(−1)(−1)=sin(1)Since, f(x) is continuous at x=0⇒limx→0f(x)=f(0)⇒sin(1)=α

Let $[x]$ denotes the greatest integer less than or equal to $x$ . Then, the value of $α$ for which the function $f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} \frac{sin [- x^{2}]}{[- x^{2}]} & x \neq 0 α & x = 0 \end{matrix}$ is continuous at $x = 0$ , is