Find the limit

Question

lim (x --> ∞) √(a² + ax) - √(x² + bx)

 

 

I can see what happens when I use a graphing calculator but I'm not allowed to on the test.

 

ANSWER: 1/2 (a - b)

#27, p141

Answer 1

 

Hi Mia,

 

If the problem is to find lim_x→∞{√(x²+ax) - √(x²+bx)},

 

then we can slove it as follows.

 

lim_x→∞{√(x²+ax) - √(x²+bx)}

 

            = lim_x→∞{√(x²+ax) - √(x²+bx)}·{√(x²+ax) + √(x²+bx)}/{√(x²+ax) + √(x²+bx)}

 

            = lim_x→∞{(x²+ax) - (x²+bx)}/{√(x²+ax) + √(x²+bx)}

 

            = lim_x→∞(a-b)x/{√(x²+ax) + √(x²+bx)}

 

            = lim_x→∞(a-b)/{√(1+a/x) + √(1+b/x)}  (both numerator & denominator divided by x)

 

            = (a - b)/2