Solving limits without hopital rule

Question

Hi, I need help understanding how to solve for: 

 

a) lim as x tends to 0 (sin|x|)/(x) 

 

b) lim as x tends to 0 (sinx)^3/ x^2

Answer 1

In class, you probably (hopefully!) learned that lim_x→0[sinx/x] = 1

 

(a).  lxl = x if x > 0    lxl = -x if x < 0

 

       To evaluate the given limit, consider the "one-sided" limits:

 

         lim_x→0^- [sinlxl/x] = lim_x→0^- [sin(-x)/x]

 

                                 = -lim_x→0^- [sinx/x]  (Recall:  sin(-x) = -sinx)

 

                                 = -1

 

         lim_x→0⁺ [sinlxl/x] = lim_x→0⁺ [sinx/x] = 1

 

         Since the one-sided limits are not the same, the given limit does          not exist.

 

(b).  lim_x→0 [sin³x/x²]

 

          = (lim_x→0 [sinx/x])(lim_x→0 [sinx/x])(lim_x→0 [sinx]) = (1)(1)(0)

 

                                                                               = 0