How do you find the limit algebraically for this problem?

Question

lim     (x-4)/((√x) -2)

x→4⁺

Answer 1

Try working with the difference of squares:  a² - b² = (a + b)(a - b)

 

x - 4 = (√x + 2)(√x - 2) ⇒

 

(x - 4) / (√x - 2) = [(√x + 2)(√x - 2)] / (√x - 2)

 

Note that (√x - 2) cancels out.

 

(x - 4) / (√x - 2) = √x + 2

 

Now you can plug in x = 4.

Answer 2

This the limit as the function approaches x=4 from the right.

 

 

First, find the vertical asymptotes and restrictions, if any, by setting the denominator equal to zero.

 

√(x) - 2 = 0

 

√x = 2

 

x = 4    , including negative values of x.  So your domain of the function is in the interval

 

(0, 4)∪(4, ∞)

 

 

Next, just evaluate the function at  x=6 , x=5 , x = 4.5, and x=4.25  in this order of x values.

 

As you evaluate the function at these x values, do the y values increase infinity, decrease infinity, or reach a certain constant number?  The answer to that question will be your limit.