Asymptotes of Functions

Question

a) If a function has a horizontal asymptote, the function's graph can never cross it

TRUE OR FALSE

b) A function can have different horizontal asymptotes at  +∞ and  −∞.  

TRUE OR FALSE

c) A function can have infinitely many different vertical asymptotes.   

TRUE OR FALSE

Answer 1

a) FALSE

For example, the function y = e^-xsin(x) has a horizontal asymptote y = 0,

yet the graph crosses y=0 an infinite number of times.

b) TRUE

For example, y = arctan(x) has horizontal asymptote π/2 at +∞, and -π/2 at -∞.

c) TRUE

For example, y = tan(x) has vertical asymptotes at x = π/2 + kπ for any integer k.