limit

[ozlem]

How can i show that both limits exist and are equal to 0? Since i couldn't find out how i could attach the image of the problems, here's the web address of the limits where i attached as jpeg file:

http://www.mathhelpforum.com/math-help/ ... 75478.html

 

[tkhunny]

What's the point of asking the same question. You have hints. Try them? What's wrong with L'hospital? Why does it matter if the derivative it difficult? Let's see your work.

 

[ozlem]

Because i want to validate my solution, and i want to learn how to solve it properly. If you can't see the point, rather than solving you can reference good examples similar to these limits

 

[tkhunny]

You're just not going to show any effort, are you? :evil:

 

[lookagain]

ozlem,

$\displaystyle what \ if \ you \ let \ log(n) = x^2 \ ?$

$\displaystyle Then \ 10^{log(n)} \ = \ 10^{x^2}$

$\displaystyle So \ n \ = \ 10^{x^2}$


$\displaystyle Suppose \ x \ = \ \sqrt{log(n)}$

$\displaystyle As \ n \ \longrightarrow \infty, \ log(n) \ \longrightarrow \infty. \ And \ then \ \sqrt{log(n)} \longrightarrow \infty.$

$\displaystyle So \ then \ x \ \longrightarrow \infty.$


As an example, look at transforming your second problem:


$\displaystyle \lim_{x \to \infty} \frac{2^x}{10^{x^2}} \ = \ \lim_{x \to \infty} \frac{2^x}{10^{x(x)}} \ = \ \lim_{x \to \infty} \frac{(2)^x}{(10^x)^x} \ = \ \lim_{x \to \infty}\bigg(\frac{2}{10^x}\bigg)^x$


What can you show/state after this?

 

[ozlem]

Thanks for the example, inside of the parathesis is 2 divided by infinity which is zero. zero to the power of infinity is zero

The first one is solved by L'Hopital which is taking derivative twice