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All that said, I have been tripped up by finding limits by conjugates.

1. The problem statement, all variables and given/known data

1. The problem statement, all variables and given/known data

Rationalize the following expression by conjugates: $$ \lim_{x \rightarrow -1} \frac { (x+1) } { \sqrt{x+5} - 2}$$

**2. Relevant equations**

Not applicable.

**3. The attempt at a solution**

So the easy way to solve this is to simply plot the function, there you clearly see that the limit is 4. However, that's too easy as this can clearly be done by hand (otherwise the exercise would have said "graph it"). The basic algebra is where I go horribly astray.

$$ \frac { (x+1) } { \sqrt{x+5} - 2} = \frac { (x+1) } { \sqrt{x+5} - 2} \frac { \sqrt{x+5} + 2 } {\sqrt{x+5} + 2} $$

Having cheated and graphed the function I can tell by inspection that the denominator should go to ## (x + 1) ## to cancel out the ## (x + 1) ## in the numerator and yield the limit of 4. And as I'm writing this I think I've answered my own question (huzzah!); it's a simple application of FOIL. $$ ( \sqrt {x + 5} - 2 ) ( \sqrt {x + 5} + 2 ) = x + 5 - 4 = (x + 1) $$ And so $$ \lim_{x \rightarrow -1} \frac { (x+1) } { \sqrt{x+5} - 2} = \lim_{x \rightarrow -1} \sqrt{x+5} + 2 = \sqrt{4} + 2 = 2 + 2 = 4 $$ Have I gone about this the right way or did I just get lucky?