regarding time -frequency domain

hello ,

i have a doubt in time frequency conversion ... why a rapidly changing signal require a large bandwidth..

why a very high frequency sine wave it self appears as a impulse at that particular frequency in frequency domain ,since it is rapidly changing with respect to time it should occupy a high band width but instead its representation is only a impulse in frequency domain.

and the other doubt is , consider a sine wave of fixed frequency but varying amplitude as a function of t. for this how the spectrum in frequency domain look like

i consider tha answer to be for the last question as a impulse at that particular frequency whose amplitude varies