1. ## question from Alta

I'm doing a homework problem that is stated: "Suppose that a is a positive constant and that R is the region bounded above by y = 1/xa , below by y = 0 and on the left by the line x = 1. Part A asks to sketch the graphs for a = .5, 1 , and 2 and then asks which is closest to the x axis.

Computationally, I know the answer is 1/x2 but is there any other way to prove that that function approaches the x axis faster than the other two functions? I attempted the limits for each function and naturally they all approach 0 when x -> infinity.

Thanks again,
Alta

2. ## Re: question from Alta

Look at the ratio $\Large \dfrac{\frac {1}{x^a}}{\frac {1}{x^b}}$ for $x>1,~a>b$

3. ## Re: question from Alta

Originally Posted by romsek
I'm doing a homework problem that is stated: "Suppose that a is a positive constant and that R is the region bounded above by y = 1/xa , below by y = 0 and on the left by the line x = 1. Part A asks to sketch the graphs for a = .5, 1 , and 2 and then asks which is closest to the x axis.

Computationally, I know the answer is 1/x2 but is there any other way to prove that that function approaches the x axis faster than the other two functions? I attempted the limits for each function and naturally they all approach 0 when x -> infinity. Alta
Here are some extremely important facts: Suppose that $0<a <1<b~\&~n\in\mathbb{Z}^+$ then:
$\large 0<a^n<a<\sqrt[n]{a}<1<\sqrt[n]{b}<b<b^n$.
If you can remember these then your homework will be easier.

Here is a favorite exercise: Given that $0<\sqrt a<\dfrac{1}{\sqrt b}<1$ then sort the following into ascending order.
$\large\sqrt{a},~\sqrt{b},~\dfrac{1}{a},~\dfrac{1} { b},~a,~b,~a^2.~\&~b^2$

If one does the problem, life is a lot easier.

thank you!

5. ## Re: question from Alta

wow, that really helped. i already intuitively understood that even though I wasn't aware of it.