Answer:
the vertical asymptotes of f(x) are (π/2, -π/2)
Step-by-step explanation:
since tan(x)= sin (x) / cos(x), we can rewrite the function as
f(x) = 4*x*sin (x) / cos(x) , for -π/2>x>π/2
since cos(π/2)= 0 , cos(-π/2)=0 and the cosine function is continuous (if we get closer to π/2 and -π/2 we will get closer to 0) , when cos(x) goes smaller, f(x) goes bigger. in the limit , when cos(x) goes to 0 , f(x) will go to infinity , therefore x=π/2 and x=-π/2 are asymptotes of f(x)
Note:
strictly speaking we say that f(x) has vertical asymptotes in x=π/2 and x=-π/2 because
when x→ π/2, lim f(x)=∞
when x→ -π/2, lim f(x)=∞
where lim is called limit of the function
