Question

Could someone help with this limit problem?

lim x-->1 of
((sqrt(6-x))-2)/((sqrt(10-x)-3))

I tried to rationalize the denominator and numerator but it didn't work!

Answer 1

Step 1 in any limit question is to actually sub in the approach value into the expression.
If you get 0/0 , you got some work ahead of you
If you get (non-zero)/0 , there is no limit
If you get a real number, that's it, you are done, that is your answer.

So, let's try this

when x = 1, we get:

(√(6-1) - 2)/(√(10-1) - 3)
= ( √5 - 2)/0
which is undefined, thus there is no limit

That is why rationalizing did not get you anywhere.