Let fleft - x right -begin-cases-sin x- xneq npi 2- x-npi end-cases- where nepsilon mathbb-Z- andgleft - x right -begin-cases-x-2-1- xneq 2 3- x-2 end-cases- Then displaystyle lim-xto 0-gleft - fleft - x right - right - is013none of these

Question

Let fleft - x right -begin-cases-sin x- xneq npi  2- x-npi end-cases- where nepsilon mathbb-Z- andgleft - x right -begin-cases-x-2-1- xneq 2  3- x-2 end-cases- Then displaystyle lim-xto 0-gleft - fleft - x right - right - is013none of these

Toppr Admin · Accepted Answer

Limx-x2192-0-g-f-x-limx-x2192-0-g-sinx-limx-x2192-0-sin2x-1-0-1-1limx-x2192-0-x2212-g-f-x-limx-x2192-0-x2212-g-sinx-limx-x2192-0-x2212-sin2x-1-1

Let $f (x) = {\begin{matrix} sin x, x \neq n π 2, x = n π \end{matrix}$ , where $n ϵ Z$ and
$g (x) = {\begin{matrix} x^{2} + 1, x \neq 2 3, x = 2 \end{matrix}$ .
Then $lim x \to 0 g (f (x))$ is
$0$
$1$
$3$
none of these

$lim x \to 0^{+} g (f (x)) = lim x \to 0^{+} g (sin x) = lim x \to 0^{+} ({sin}^{2} x + 1) = 0 + 1 = 1 lim x \to 0^{-} g (f (x)) = lim x \to 0^{-} g (sin x) = lim x \to 0^{-} ({sin}^{2} x + 1) = 1$

Let f(x)={sinx,x≠nπ2,x=nπ, where nϵZ andg(x)={x2+1,x≠23,x=2. Then limx→0g(f(x)) is013none of these

limx→0+g(f(x))=limx→0+g(sinx)=limx→0+(sin2x+1)=0+1=1limx→0−g(f(x))=limx→0−g(sinx)=limx→0−(sin2x+1)=1

Let $f (x) = {\begin{matrix} sin x, x \neq n π 2, x = n π \end{matrix}$ , where $n ϵ Z$ and
$g (x) = {\begin{matrix} x^{2} + 1, x \neq 2 3, x = 2 \end{matrix}$ .
Then $lim x \to 0 g (f (x))$ is
$0$
$1$
$3$
none of these

$lim x \to 0^{+} g (f (x)) = lim x \to 0^{+} g (sin x) = lim x \to 0^{+} ({sin}^{2} x + 1) = 0 + 1 = 1 lim x \to 0^{-} g (f (x)) = lim x \to 0^{-} g (sin x) = lim x \to 0^{-} ({sin}^{2} x + 1) = 1$