Help on a few L'Hopital's Rule Problems!?

Question

1) Find the Limit as x ->infinity of (1 + (10/x))^4x

I've gotten tot he point where it is a 1 ^ Infinity form and have used the Natural Log properties to get it to Ln y = 4x Ln(1 + (10/x))

I do not know where to go from there

2) Find the Limit as x--> Infinity of (x e^(1/x) - x)

I believe this is an Infinity - Infinity problem but don't know how to get it into a 0/0 or Infinity/Infinity form.

3) Find the Limit as x ---> positive Infinity of Sqrt(2+ (x^2)) / x

(Just incase, Sqrt = Square Root)


Any help or suggestions would be great

Thanks

Answer 1

It's perhaps counterintuitive, but write

4x Ln(1 + (10/x))

as

Ln(1 + (10/x))/(1/4x)

For (2), factor as x(e^(1/x) -1) then, similar to above, write this as

(e^(1/x) -1)/(1/x)

If you use l'Hopital's rule on (3), you get
Limit as x ---> positive Infinity of x/Sqrt(2+ (x^2))
Note that you now have the reciprocal of the original limit.

Or just take the dividing x into the sqrt and evaluate
Limit as x ---> positive Infinity of Sqrt((2/x²) + 1)

Answer 2

For 1) if you know what happens to ( 1 + 1/n)^n as n tends to infinity (and you should) then let 1/n = 10/x......x= 10n and use your knowledge of exponents then the answer is apparent. Note the answer is HUGE but finite. Otherwise Ron did a good job.