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

Algebra ->  Radicals -> 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      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
If we take the direct route, we get:


lim%28x-%3E7%2C%28sqrt%28x%2B2%29-3%29%2F%28x-7%29%29 Start with the given limit.


%28sqrt%287%2B2%29-3%29%2F%287-7%29 Plug in x=7 (let's just pretend that we can evaluate this limit the intuitive way).


%28sqrt%289%29-3%29%2F%287-7%29 Add


%283-3%29%2F%287-7%29 Take the square root of 9 to get 3.


%280%29%2F%280%29 Subtract


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

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


lim%28x-%3E7%2C%28sqrt%28x%2B2%29-3%29%2F%28x-7%29%29 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 sqrt%28x%2B2%29-3


lim%28x-%3E7%2C%28%28x%2B2%29-9%29%2F%28%28x-7%29%28sqrt%28x%2B2%29%2B3%29%29%29 FOIL the numerator (use the difference of squares formula).


lim%28x-%3E7%2C%28x-7%29%2F%28%28x-7%29%28sqrt%28x%2B2%29%2B3%29%29%29 Combine like terms.


lim%28x-%3E7%2Ccross%28%28x-7%29%29%2F%28cross%28%28x-7%29%29%28sqrt%28x%2B2%29%2B3%29%29%29 Cancel out the common factors.


lim%28x-%3E7%2C1%2F%28sqrt%28x%2B2%29%2B3%29%29 Simplify. Now let's see if we can evaluate this limit using the naive approach (ie use lim%28x-%3Ea%2Cf%28x%29%29=f%28a%29).


1%2F%28sqrt%287%2B2%29%2B3%29 Plug in x=7


1%2F%28sqrt%289%29%2B3%29 Add


1%2F%283%2B3%29 Take the square root of 9 to get 3.


1%2F6 Add


So lim%28x-%3E7%2C%28sqrt%28x%2B2%29-3%29%2F%28x-7%29%29=1%2F6


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).