L'Hopital and 0 X Infinity

[onedayatatime]

Hello,

I am curious about how to do this problem # 8 attached in the screenshot. This problem is attached to a broader worksheet about L'Hopital's rule so I am assuming I have to use the rule to solve the problem.

My guess is that I have to turn this into rational form by putting one of the terms to the negative -1 power. There is another problem on the worksheet that also asks you to prove that 0 X ∞ is equivalent to ∞/∞. How could this be so?

Thank you!

 

[Dr.Peterson]

Suppose that f(x) approaches infinity and g(x) approaches 0. Then 1/f(x) approaches 0, so g(x)/f(x) has the form 0/0. Similarly, 1/g(x) approaches infinity, so f(x)/g(x) has the form infinity/infinity. I imagine one of those is what you did for the three specific examples.

That's essentially what you described, but considerably more specific.

 

[Steven G]

Teachers do a great job in showing students how to change a division problem into a multiplication problem. However, they also need to show how to change a multiplication problem into a division problem.
The rule is simple: Keep the 1st factor the same, change multiplication to division and then divide by the reciprocal of the 2nd factor.

[math] ab = \dfrac{a}{\frac{1}{b}}[/math]