i need help with this question please.
The functions f(x) and g(x) are defined below as:
f(x)=sinh(x)= [e^x- e^(-x)]/2 and g(x)= 1/(1+x^2 ) for all real x
Show that f(x)=g(x) has a unique solution for x in (-∞,∞).
Thanks for any help.
Raoh is right. apply the theorem to the function h(x) = f(x) - g(x). show that h(x) = 0 for some x. (showing that h(x) = 0 is equivalent to showing f(x) = g(x), hope you see that) all you need to do is find two x-values where h(x) changes sign and apply the theorem (be sure to check all the hypotheses)
you were never asked to find a solution (your solution would only be an approximation, by the way), you were simply asked to show one exists. do this by applying the intermediate value theorem as i said and as you did in this thread.
you don't have to use infinity. you can pick any two numbers you wish, as long as the functions change sign for those values. you can then apply the theorem on some closed interval containing those values.
if you really want to incorporate infinity, you'd have to use limits.