Evaluate the limit, if it exists.

Question

lim

x-->3    (1/x-1/3)/(x-3)

Answer 1

Recall that f'(a) = lim_x→a[(f(x) - f(a))/(x-a)]

 

Letting a = 3 and f(x) = 1/x, the given limit is equal to f'(3).

 

Since f'(x) = -1/x², we have f'(3) = -1/9.  

Answer 2

You have what looks like an indeterminate form.  In this case, it looks like 0/0.  If possible, simplify the expression algebraically, and you might get a form that is not indeterminate.  In this problem,

 

(1/x - 1/3) / (x - 3) = (3x / 3x)[(1/x - 1/3) / (x - 3)] = (3 - x) / [3x(x - 3)]

 

= -(x - 3) / [3x(x - 3)] = - 1 / 3x

 

This is not an indeterminate form, and you can substitute x = 3 and get -1/9.  However, if you are left with and indeterminate form, you could use L'Hopital's rule, but that is not necessary in this problem

Answer 3

Evaluate the expression at x=2.99999 and at x=3.00001.  If the results are closely the same, then there is a limit.  If they the results far from the same, then the limit does not exist.

Answer 4

lim_{x-->3} (1/x-1/3)/(x-3)=lim_{x-->3} (3-x)/3x)/(x-3)=-lim_{x-->3}(1/3x)=-1/9