Question

How do you evaluate the limit `(sqrt(x+6)-x)/(x-3)` as x approaches `3`?

Answer 1

`-5/6`

Explanation:

As the function is indeterminate `0/0` when x = 3

Multiply the numerator/denominator by the conjugate of the numerator.

`sqrt(x+6)-x" conjugate "tosqrt(x+6)+x`

`rArr((sqrt(x+6)-x)(sqrt(x+6)+x))/((x-3)(sqrt(x+6)+x)`

`=(x+6-x^2)/((x-3)(sqrt(x+6)+x))`

`=(-cancel((x-3))(x+2))/(cancel((x-3))(sqrt(x+6)+x))`

exclusion x ≠ 3

`=(-(x+2))/(sqrt(x+6)+x)`

`rArrlim_(xto3)(sqrt(x+6)-x)/(x-3)`

`=lim_(xto3)(-(x+2))/(sqrt(x+6)+x)=-5/6`