Re: prove or desprove claims

Quote:

Originally Posted by

**transgalactic** [LEFT]

determine if the following are correct or wrong

if correct prove if not then show counter example:

a)if $\displaystyle lim_{x->\infty}f(x)=\infty$ and $\displaystyle g(x)>0$ for $\displaystyle x\in R$ then

$\displaystyle lim_{x->\infty}f(x)g(x)=\infty$

couldnt find counter example ,i thought f(x)=x but there is not g(x)

to desprove it

(a) how about $\displaystyle f(x) = x$ and $\displaystyle g(x) = \frac{1}{x^2+1}$ for a counter-example?

Re: prove or desprove claims

(a)

I like your idea. How about $\displaystyle f(x)=x, g(x)=1/x^2$ ?

(d)

is the following function continuous? $\displaystyle g(c) = \int_a^c f(x) dx$

if so you can use the intermediate value theorum, as its trivial to show that there are values of g(c) on either side of 1.

Re: prove or desprove claims

yes thanks :)

1 down 3 to go

Re: prove or desprove claims

regarding b) i was told to turn $\displaystyle 2xArctanx>ln(1+x^{2})$ into $\displaystyle 2xArctanx-ln(1+x^{2})>0$

$\displaystyle f(x)=2xArctanx-ln(1+x^{2})$

so i need to prove that f(x)>0 for x>0

f(0)=0

now i need to show that f'>0 so f will be monotonicly increasing

$\displaystyle f'(x)=2Arctanx+2\frac{1}{1+x^2}- \frac{2x}{1+x^{2}}$

how to conclude that f' is positive for x>0