limit invoving radicals

[lmerry]

Hi there,

My roommate has this problem for his Calc 1 homework. I'm currently taking Calc 2 and it's driving me crazy that i can't get it. I know im just forgetting some little thing, but it would be great if someone could help out.

\(\displaystyle \lim_{x\to\infty}\sqrt{x^2+ax}-\sqrt{x^2-bx}.\)

Again, its a fairly simple problem and i guess i need to brush up on my algebra!

Thanks in advance,

Luke

 

[galactus]

Hello:

Multiply the top and bottom by the conjugate:

\(\displaystyle \L\\\lim_{x\to\infty}\frac{(\sqrt{x^{2}+ax}-\sqrt{x^{2}-bx})}{1}\cdot\frac{(\sqrt{x^{2}+ax}+\sqrt{x^{2}-bx})}{(\sqrt{x^{2}+ax}+\sqrt{x^{2}-bx})}\)

\(\displaystyle \L\\=\lim_{x\to\infty}\frac{(a+b)x}{\sqrt{x^{2}+ax}+\sqrt{x^{2}-bx}}\)

Divide top by x and bottom by \(\displaystyle \sqrt{x^{2}}\)

This gives:

\(\displaystyle \L\\\lim_{x\to\infty}\frac{a+b}{\sqrt{1+\frac{a}{x}}+\sqrt{1-\frac{b}{x}}}\)

Now, see the limit?.

 

[lmerry]

yep, thanks a lot. I was really close. I was going back and fourth, multiplying by the conjugate 3 or 4 times..heh.

This is good practice for me as well, because i have an exam next week that covers improper integrals, and i need to be able to manipulate limits that approach infinity.