SOLUTION: Evaluate each limit. Rationalize the numerator by multiplying both numerator and denominator √x+1.
lim
x→1
x-1/√x-1
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Question 273224: Evaluate each limit. Rationalize the numerator by multiplying both numerator and denominator √x+1.
lim
x→1
x-1/√x-1
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
The "trick" you will see here is used a lot to find limits where the denominator appears to approach zero. What we will do is multiply by its conjugate, . From the pattern we know that when you multiply conjugates you get the difference of the squares of the two terms. Let's see how this helps:
which simplifies to:
A fine but important point is that in this limit x approaches 1 but is never actually equal to 1! This is important because if x is 1, x-1 is zero and we cannot cancel 0/0. But since x is never 1, x-1 is never 0 and we can cancel the (x-1)'s:
This limit is simple to find. I'll leave it up to you to finish.
Responding to your message: This is all correct as long as the problem you posted is correct. Of course if the problem is actually:
then your answer will be "upside down", too (which matches the answer key's answer of 1/2).
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