f(x)=(2x²)/(x+1)
for vertical asymptotes we look at where the demoninator is 0. here that means x = -1
f(-1)=(2(-1)²)/(-1+1) = 2/oo
if we explore x = -1 + h with 0 < abs h "<<" 1 we have
f(h-1)=(2(h-1)²)/(h)
the numerator is always positive so
if h < 0, ie the left-sided limit, then the limit is -oo
if h > 0, ie the right-sided limit, then the limit is +oo
so
lim_{x to -1 ^ -} f(x) =- oo
lim_{x to -1 ^ +} f(x) =+ oo
for horixontal and oblique asymptotes we look at x to pm oo
here, lim_{x to pm oo }(2x²)/(x+1) = pm infty as the quadratic power dominates the expression
but note also that
lim_{x to pm oo }(2x²)/(x+1) = lim_{x to pm oo }(2x)/(1+1/x) approx lim_{x to pm oo }(2x)/(1) as the 1/x term diminishes in the denominator
so the line y = 2x is an oblique asymptote