<p>Actually, 314159265 is right. L’Hopital’s Rule is not required for any limits on the AP Calculus AB exam. It is an AP Calculus BC topic exclusively.</p>
<p>For the limit as x -> infinity of x<em>sin(1/x), you have to recall that the limit as x approaches 0 of sin(x) / x = 1. (This is a limit that many instructors have students memorize early in the year, as you use it later on to prove that the derivative of sin x = cos x.) The trick is to re-write x as 1/(1/x). Then the lim x-> infinity of x</em>sin(1/x) = lim x->infinity of sin(1/x) / (1/x). Let u = 1/x. Then as x->infinity, u->0, so lim x->infinity of sin(1/x) / (1/x) = lim u-> 0 of sin(u)/u, which we established earlier as 1.</p>