How do you find limit of ((√(x+2)-3)/(x-7)) as (x→7)⁡?

Answer 1

[x→7]lim (√(x+2) - 3)/(x-7)

First, multiply both numerator and denominator by √(x+2) + 3:

[x→7]lim (x+2 - 9)/((x-7)(√(x+2) + 3))

Simplify:

[x→7]lim (x - 7)/((x-7)(√(x+2) + 3))
[x→7]lim 1/(√(x+2) + 3)

Evaluate:

1/(√(7+2) + 3)
1/6

And we are done.

Answer 2

Either take the derivative of the top and the bottom separately, in which case you get [(1/2)(x+2)^(-1/2)]/1; if you plug in x=7 you will get 1/6...

or you can plug 6.99999 and 7.00001 into a calculator and see if those numbers are nearly the same; if they are, then you can pretty much approximate the limit.

Either way, you will arrive at 1/6.