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

1 Answer
Steve M
Nov 23, 2016

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]}

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