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Find the limit and prove that exists lim (x,y) approach (0,0) (sqrt(x^2y^2+1) - 1) / (x^2 + y^2)

Please explain how to do this problem.

Here's the problem, n=1 to infinity (5^k /(3^k + 4^k).

So I know that you can use the test for convergence to figure this out, but at first glance, I used the root test and I got the answer equal to 0 meaning it would be convergent.

So by doing the root test, I get lim n(to infinity) 5/ sqrt (3^k + 4^k)

I know the limit of this is 0 do to the stable numerator vs the infinitally increasing denominater.

So can someone explain to me where I went wrong when doing the root test because the series itself is a divergent but I got a convergent using the root test.

a job that pays 9,000 dollars a month or

a job that pays 1 penny the first day, 2 pennies the second, 4, 16, etc. So n^2 where n= number of day.

Which will it be?

1. n=1 to infinity ( n! / n^n)

Please show steps and explain thoroughly.

What's one thing you would want to become much more valuable (perhaps more than gold)?