Answer to Question #236202 in Math for Fia

Question

Determine the Vertical Asymptote

f(x) = 1-x / x^3 -64

f(x) = 7x + 2/ x^2+1

Answer 1

1.

$f(x)=\dfrac{1-x}{x^3-64}, x\in \R$

$x^3-64\not=0$

$(x-4)(x^2+x+16)\not=0$

$x\not=4$

$\lim\limits_{x\to4^-}f(x)=\lim\limits_{x\to4^-}\dfrac{1-x}{x^3-64}$

$=\lim\limits_{x\to4^-}\dfrac{1-x}{(x-4)(x^2+x+16)}=\infin$

$\lim\limits_{x\to4^+}f(x)=\lim\limits_{x\to4^+}\dfrac{1-x}{x^3-64}$

$=\lim\limits_{x\to4^+}\dfrac{1-x}{(x-4)(x^2+x+16)}=-\infin$

Vertical asympote: $x=4.$

2.

$f(x)=\dfrac{7x+2}{x^2+1}, x\in \R$

$x^2+1>0, x\in\R$

There is no vertical asymptote.