Question

Asymptote(s) of the function, $f\left(x\right)=\frac{x^{-x}-2}{x-2}$ is/are

• A. $x=2$
• B. $x=1$
• C. $y=x+1$
• D. $y=x-1$

Answer 1

Answer: (A), (C)

$x=2$
$y=x+1$

$y=(\frac{x^2-2}{x-2})$

for vertical asymotote ,find the point where denominator equal to $0$

hence $x=2$

now put them on equ(1),we will get $y=\infty$

hence $x=2$ is a vertical asymptote

$horizontal$ $asymptote$

line $y=L$ is a horizontal asymptote of fuction if either ${}_{x\to \infty }^{lim}f\left(x\right)=L$ and ${}_{x\to -\infty }^{lim}f\left(x\right)=L$

where $L$ is finite number

${}_{x{\to }_{-}^{+}\infty }^{lim}(\frac{x^2-2}{x-2}){=}_{-}^{+}\infty$

hence there is no horizonatal asymptote .

$slant$ $asymptote$

do polynomial long division ,

$y=(\frac{x^3-27}{2x^2-4})=x+2+\frac{2}{x-2}$

the rational polynomial approaches $0$ as the variable approaches infinity

hence $y=x+2$ is a slant asymptote