Evaluating the limit of (x^3 + 3x^2)^(1/3) - (x^2 - 2x)^(1/2) as x -> infinity

[bgstudent123]

Hey everybody!

I need to find the limit of the following function as x goes to positive infinity.

The function is:

. . . . .\(\displaystyle f(x)\, =\, \sqrt[3]{\strut x^3\, +\, 3x^2\,}\, -\, \sqrt{\strut x^2\, -\, 2x\,}\)

I tried solving it, but have no real progress. I did start by trying to rationalize everything, but that didn't take me far.

If someone could help me, I would be very grateful!

Thanks in advance!
bgstudent123

 

[tkhunny]

I need to find the limit of the following function as x goes to positive infinity.

The function is:

. . . . .\(\displaystyle f(x)\, =\, \sqrt[3]{\strut x^3\, +\, 3x^2\,}\, -\, \sqrt{\strut x^2\, -\, 2x\,}\)

I tried solving it, but have no real progress. I did start by trying to rationalize everything, but that didn't take me far.

Have you considered exploring \(\displaystyle e^{f(x)}\)?

Note "x goes to positive infinity" is a very awkward statement. Infinity isn't a place. Try something like "x increases without bound in the positive direction".

 

[stapel]

I need to find the limit:

. . . . .\(\displaystyle \displaystyle \lim_{x \rightarrow \infty}\, \sqrt[3]{\strut x^3\, +\, 3x^2\,}\, -\, \sqrt{\strut x^2\, -\, 2x\,}\)

I tried solving it, but have no real progress. I did start by trying to rationalize everything, but that didn't take me far.

There are no fractions in the original expression, so I don't know what you mean when you refer to "rationalizing everything"...?

What tools, theorems, algorithms, etc, were recently covered in class? What was the subject of the section in your textbook from which this exercise came?

When you reply, please show your work, so we can see what you're doing. Thank you!