Limit Exercise, as x Approaches Infinity

[noreen1]

Hi, I am struggling with this same problem no 24 but I can't see how you worked it out? Can you show me?

i hope this helps with 24.

24. \(\displaystyle \; \lim\limits_{x \to \infty} \sqrt{x^2+ax} - \sqrt{x^2+bx}\)

 

[Quaid]

Hi Noreen:

Where are you stuck? Did you try anything?

 

[HallsofIvy]

A good way of handling a difference of square roots is to multiply and divide by the sum of the square roots:\(\displaystyle \sqrt{u}- \sqrt{v}= \left(\sqrt{u}-\sqrt{v}\right)\frac{\sqrt{u}+ \sqrt{v}}{\sqrt{u}+ \sqrt{v}}= \frac{u- v}{\sqrt{u}+ \sqrt{v}}\)

After you have done that, in problems involving x going to infinity, it is often a good idea to divide both numerator and denominator by x so to get x in the denominators of one or more fractions. As x goes to infinity, \(\displaystyle \frac{1}{x}\) goes to 0.