Evaluate each limit, if it exsists.

Question

Here are some examples of questions in the textbook that I don't know how to do for my test!

 

Evaluate each limit:

 

lim (x approaches 0) tanx/x

 

 

lim (x approaches 0) sin6x/sin8x

 

 

lim (x approaches 0) sinx/square root of x

 

 

lim (x approaches 0) xsinx/1-cosx

Answer 1

lim (x->0) sin(6x)/sin(8x) = lim (x->0) (6 cos(6x))/(8 cos(8x))  (Using L'Hopitals Rule)

                                  = (6 * 1)(8 * 1) = 6/8 = 3/4

 

lim (x-0) sin(x)/x^(1/2)  = lim  (x->0) cos(x)/(1/2 * x^(-1/2) (Using L'Hopitals Rule)

                                 = lim (x->0) 2 *x^(1/2) * cos(x) = 2 * 0 * 1 = 0

 

lim (x * sin(x)) /(1 - cos(x)) = lim (x->0) (sin(x) + x * cos(x))/sin(x) (Using L'Hopitals Rule)

                                      = lim (x->0) (cos(X) + cos(x) - x *sin(x))/cos(x) (Using L'Hopitals Rule)

                                      = lim (x->0) (2 - x*sin(x)/cox(x)) = 2

Answer 2

Use L'hopitals rule for each one. 

 

With L'hopitals, you will take a derivative of the top independent of the bottom.  And take a derivative of the bottom independent of the top. 

 

lim            tanx/x

x->0

 

Take derivative of tan x which is sec²x.  Then take a derivative of the bottom which is 1.  So you will get sec²x/1.  After this plug in 0.  Sec is 1/cos.  Cos of 0 is 1.  So you will get 1/1.  So your answer is 1. Do the same with the rest.