State T/F with proper justification -1< tanhx<1 and -infinity<= tanx<=infinity?

Answer 1

tan x=sin x/cosx
and
(sin x)^2+(cos x)^2=1

highest value of sin x=1

than, if sin x=1, cos x=0

so tan x=1/0=infinity

again you can see, if sin x=-1, cos x=0 and then tan x= -infinity.

and if sin x= y, cos x=+-(1+y^2)^.5 [condition: IyI<1]

using binomial law you can easily show that
y/+-(1+y^2)^.5 can be resolved in an infinite series which can take any real value depending upon the value of y.

so easily
-1< tanhx<1 is false.
-infinity<= tanx<=infinity is true

Answer 2

tan(x) = sin(x) / cos(x)... lim when x->pi (sin(x) / cos(x)) = lim x->0 ((x + 1)/ x) = infinity. also you get the min value for x->3/2 pi. also the same for tanhx (formula should be something with (e^x+e^-x)/2, don't rember correctly), and use limit formula with e^x (there is one, which i also don't rember)

Answer 3

hmmmmmm