Question

The graphical representation of the curve $y=\frac{x}{x^2-4}$ is

• A.
• B.
• C.
• D. None of the these

Answer 1

The correct option is A

$y=\frac{x}{x^2-4}\Rightarrow$ Domain is $R-\{\pm 2\}$ and $y$ is odd function.
As $x\to \pm \infty ,y\to 0$ and
As $x\to 2^{+},y\to +\infty$ and $x\to 2^{-}\Rightarrow y\to -\infty$
As $x\to -2^{+},y\to +\infty$ and $x\to -2^{-}\Rightarrow y\to -\infty$
Also, $y\left(0\right)=0$
Now $y=\frac{x}{x^2-4}\Rightarrow \frac{dy}{dx}=\frac{-(x^2+4)}{{(x^2-4)}^2}<0\forall x\in R-\{\pm 2\}$
$y=\frac{x}{(x+2)(x-2)}\Rightarrow y<0\forall x\in (-\infty ,-2)\cup (0,2)$
and $y>0\forall x\in (-2,0)\cup (2,\infty )$
The graph can be obtained as: