Would like Help with Calc problem!

**MisanthropicMan** · August 1st 2010, 07:51 PM

Here's the problem
limit (sqrt(x^2+5x)-x)
x->[infinity]

The answer is (according to two websites and my calculator) 5/2.

Here's the work I've done so far.

[Multiply the conjugate]
(sqrt(x^2+5x)-x)/1 * (sqrt(x^2+5x)+x)/(sqrt(x^2+5x)+x)

x^2+5x-x^2/(sqrt(x^2+5x)+x) = 5x/(sqrt(x^2+5x)+x)

and I don't know how to get farther than that. Please help!

**Gusbob** · August 1st 2010, 08:36 PM

Originally Posted by MisanthropicMan

Here's the problem
limit (sqrt(x^2+5x)-x)
x->[infinity]

The answer is (according to two websites and my calculator) 5/2.

Here's the work I've done so far.

[Multiply the conjugate]
(sqrt(x^2+5x)-x)/1 * (sqrt(x^2+5x)+x)/(sqrt(x^2+5x)+x)

x^2+5x-x^2/(sqrt(x^2+5x)+x) = 5x/(sqrt(x^2+5x)+x)

and I don't know how to get farther than that. Please help!

Continuing from where you left off:
$ \displaystyle\lim_{x\to\infty} \frac{5x}{\sqrt{x^2+5x}+x}$

Here is a nice little trick: factor out an x from the denominator:

$= \displaystyle\lim_{x\to\infty} \frac{5x}{x\left(\frac{\sqrt{x^2+5x}}{x} + 1\right)}$

$\displaystyle\lim_{x\to\infty} = \frac{5}{\frac{\sqrt{x^2+5x}}{x} + 1}$

$=\displaystyle\lim_{x\to\infty} \frac{5}{\sqrt{\frac{x^2+5x}{x^2}} + 1}$

As everything else is constant:

$= \frac{5}{\sqrt{\left(\displaystyle\lim_{x\to\infty }\frac{x^2+5x}{x^2}\right)} + 1}$

I trust you can continue from here?

**HallsofIvy** · August 2nd 2010, 01:53 AM

Originally Posted by Gusbob

Continuing from where you left off:
$ \displaystyle\lim_{x\to\infty} \frac{5x}{\sqrt{x^2+5x}+x}$

Here is a nice little trick: factor out an x from the denominator:

$= \displaystyle\lim_{x\to\infty} \frac{5x}{x\left(\frac{\sqrt{x^2+5x}}{x} + 1\right)}$

Or, same thing, divide both numerator and denominator by x.

$\displaystyle\lim_{x\to\infty} = \frac{5}{\frac{\sqrt{x^2+5x}}{x} + 1}$

$=\displaystyle\lim_{x\to\infty} \frac{5}{\sqrt{\frac{x^2+5x}{x^2}} + 1}$

As everything else is constant:

$= \frac{5}{\sqrt{\left(\displaystyle\lim_{x\to\infty }\frac{x^2+5x}{x^2}\right)} + 1}$

I trust you can continue from here?

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