Question

Given the function

Answer 1

$lim \: f(x) = ( \infty + 4)( \infty - 2) {}^{2} \\ x - > \infty$

$lim \: f(x) = \infty \times \infty \\ x - > \infty$

$lim \: f(x)= \infty \\ x - > \infty$

$lim \: \frac{f(x)}{x} = \frac{x(1 - \frac{4}{x})(x - 2) {}^{2} }{x} \\ x - > \infty$

$lim \: \frac{f(x)}{x} = (1)( \infty - 2) {}^{2} \\ x - > \infty$

$lim \: \frac{f(x)}{x} = \infty \\ x - > \infty$

We can then say that the function f(x)=(x-4)(x-2)² admits an asymptotic direction parallel to the y-axis at +∞ and -∞ as well since we have to follow the same steps.