SOLUTION: I can't figure out how to simplify the expression (root x+2) - 3/(x-7). I'm actually in calculus I, and I have to take the limit of this expression as x goes to 7. I know I need to

Question 219650: I can't figure out how to simplify the expression (root x+2) - 3/(x-7). I'm actually in calculus I, and I have to take the limit of this expression as x goes to 7. I know I need to factor, but I don't know where to start. Any help would be very appreciated!
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
If we take the direct route, we get:

Start with the given limit.

Plug in (let's just pretend that we can evaluate this limit the intuitive way).

Add

Take the square root of 9 to get 3.

Subtract

Since the above is indeterminate, we have to take another approach...

-------------------------------------------------

Start with the given limit.

Because there's a radical in the numerator, we can try to rationalize the numerator in hopes that something will cancel.

Multiply both the numerator and denominator by the conjugate of

FOIL the numerator (use the difference of squares formula).

Combine like terms.

Cancel out the common factors.

Simplify. Now let's see if we can evaluate this limit using the naive approach (ie use ).

Plug in

Add

Take the square root of 9 to get 3.

Add

So

An alternative (and better way in my opinion) is to use L' Hospital's Rule. If you haven't learned that yet, then ignore that last suggestion (note: the rule only works if you're dealing with interderminate forms).
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