lim  (sqrt x - sqrt 3) / (x-3) Please show work.
x->3

Hi cousinit!
We can solve this problem by applying the following identity:

(a+b)(a-b) = a^2 + ab - ab - b^2 = a^2 - b^2

where ^ means "to the power of".

The limit you want to find is the following:

lim      (sqrt x - sqrt 3) 
x->3     -----------------
              x-3

Now we can multiply both the numerator and the denominator by

(sqrt x + sqrt 3)

The idea is to get an (x-3) in the numerator that will cancel out with
that same term in the denominator. So we have:

lim      (sqrt x - sqrt 3)  (sqrt x + sqrt 3)
x->3     -----------------.-------------------
              x - 3         (sqrt x + sqrt 3)


Applying the identity I mentioned before, we have that:

(sqrt x - sqrt 3)(sqrt x + sqrt 3) = x - 3

Therefore, we get:

lim           x - 3
x->3     -----------------.-------------------
              x - 3         (sqrt x + sqrt 3)

which is the same as:

lim           1
x->3     -----------------
         (sqrt x + sqrt 3)

Finally, since the function 1/(sqrt(x)+sqrt(3)) is continuous, we can
just replace x=3 in order to get:

lim           1                   1
x->3     ----------------- =   ----------
         (sqrt x + sqrt 3)      2sqrt(3)


I hope this helps! If you have any doubts regarding my answer, please
don't hesitate to request a clarification before rating it. Otherwise
I await your rating and final comments.

Best wishes!
elmarto

Thank you so much, especially for taking the time to explain the steps in words.