How do you find the limit of # (sqrt(3x - 2) - sqrt(x + 2))/(x-2)# as x approaches 2?

The first thing we do in limit problems is substitute the #x# value in question and see what happens. Using #x=2#, we find:
#lim_(x->2)(sqrt(3x-2)-sqrt(x+2))/(x-2)=(sqrt(3(2)-2)-sqrt((2)+2))/((2)-2)=(sqrt(4)-sqrt(4))/0=0/0#

You may be wondering how this helps us. Well, because we have #0/0#, this problem becomes fair game for an application of L'Hopital's Rule. This rule says that if we evaluate a limit and get #0/0# or #oo/oo#, we can find the derivative of the numerator and denominator and try evaluating it then.

So, without further ado, let's get to it.

Derivative of Numerator
We're trying to find #d/dx (sqrt(3x-2)-sqrt(x+2))# here. Using the sum rule, we can simplify this to #d/dxsqrt(3x-2)-d/dxsqrt(x+2)#. Taking the derivative of the first term, we see:
#d/dxsqrt(3x-2)=(3x-2)^(1/2)=3/(2sqrt(3x-2))->#Using power rule and chain rule

For the second term,
#d/dxsqrt(x+2)=(x+2)^(1/2)=1/(2sqrt(x+2))->#Using power rule

Thus, our new numerator is: #3/(2sqrt(3x-2))-1/(2sqrt(x+2))#.

Derivative of Denominator
This one is fairly easy: #d/dx(x-2)=1#. Yep, that's it.

Put it all Together
Combining these two results into one, our new fraction is #(3/(2sqrt(3x-2))-1/(2sqrt(x+2)))/1=3/(2sqrt(3x-2))-1/(2sqrt(x+2))#. We can now evaluate it at #x=2# and see if anything changes:
#lim_(x->2)(3/(2sqrt(3x-2))-1/(2sqrt(x+2)))=3/(2sqrt(3(2)-2))-1/(2sqrt((2)+2))=3/(2sqrt(4))-1/(2sqrt(4))=1/2#

And there you have it! We can confirm this result by looking at the graph of #(sqrt(3x-2)-sqrt(x+2))/(x-2)#:
graph{(sqrt(3x-2)-sqrt(x+2))/(x-2) [-0.034, 3.385, -0.205, 1.504]}

#(sqrt(3x-2)-sqrt(x+2))/(x-2) = (sqrt(3x-2)-sqrt(x+2))/(x-2) *(sqrt(3x-2)+sqrt(x+2))/(sqrt(3x-2)+sqrt(x+2))#

# = ((3x-2)-(x+2))/((x-2)(sqrt(3x-2)+sqrt(x+2)))#

# = (2x-4)/((x-2)(sqrt(3x-2)+sqrt(x+2)))#

# = (2cancel((x-2)))/(cancel((x-2))(sqrt(3x-2)+sqrt(x+2)))#

So we have

#lim_(xrarr2)(sqrt(3x-2)-sqrt(x+2))/(x-2) = lim_(xrarr2)2/(sqrt(3x-2)+sqrt(x+2))#

# = 2/(sqrt(3(2)-2)+sqrt((2)+2))#

# = 2/(sqrt(4)+sqrt(4)) = 2/(2+2) = 1/2#