determining if a Signal is band limited

Hi, How can I tell if a signal is bandlimited? for example we were given the question:

x(t)=cos(500 pi t) +sin(700 pi t)

and asked if it is bandlimited and asked to find the minimum sampling rate

you should find the fourier transform for the signal and determine the bandwidth of the obtained signal (maximum frequency taken - minimum frequency). sampling rate should be 2*bandwidth

Whether the signal is band limited or not is just a question of precision. There are no band limited signal at all.
The exact band limiting of signal means its infinity in time, and vice-versa - if the signal starts and ends ever it has an infinite band.
So you should assume all signals are band limited.

The equation for harmonic signal is
x(t)= A*cos(2*pi*f*t + fi)
in your case fi = 0 and 2*f = 500 and 700 respectively. Sin signal is a shifted cosine, so the frequency doesn't change.

As already said by Fadi Mkhayel, in order to understand if a signal is bandlimited you have to find its Fourier series. The series is composed by "sin" and "cos". If the series extends to infinity then the signal is not bandlimited.
In your case you have just 2 frequencies, the the signal is band limited.

Remember that the signal is described as sin(ω•t+φ) or cos(ω•t+φ)

Since ω=2•Π•f, in your case 500•Π=2•Π•f ==> f1=250 Hz and 700•Π=2•Π•f2 ==> f2=350 Hz

from Nyquist you have to sample at a frequency greater than 2 times the higher frequency of the signal, in your case 350 Hz, than you have to sample at a frequency > 700 Hz (strictly ">" and not ">=").

In case of non-bandlimited signal you cannot find a suitable sampling frequency, than you have to filter before sampling and then choose a sampling frequency greater than the double of the cut-frequency of the filter.