Question

How do you evaluate the limit `(sqrtx-2)/(x-4)` as x approaches `4`?

Answer 1

`lim_(x rarr 4) (sqrt(x)-2)/(x-4) = 1/4`

Explanation:

The denominator can be viewed as the difference of two squares, so we can write:

`lim_(x rarr 4) (sqrt(x)-2)/(x-4) = lim_(x rarr 4) (sqrt(x)-2)/(sqrt(x)^2-2^2)`
`:. lim_(x rarr 4) (sqrt(x)-2)/(x-4) = lim_(x rarr 4) (sqrt(x)-2)/((sqrt(x)-2)(sqrt(x)+2))`
`:. lim_(x rarr 4) (sqrt(x)-2)/(x-4) = lim_(x rarr 4) 1/((sqrt(x)+2))`
`:. lim_(x rarr 4) (sqrt(x)-2)/(x-4) = 1/((sqrt(4)+2))`
`:. lim_(x rarr 4) (sqrt(x)-2)/(x-4) = 1/((2+2))`
`:. lim_(x rarr 4) (sqrt(x)-2)/(x-4) = 1/4`

This is confirmed visually from the graph `y=(sqrt(x)-2)/(x-4)`
graph{(sqrt(x)-2)/(x-4) [-0.166, 4.835, -1.01, 1.49]}