I wanted to put it as a comment but couldnt do that and I am by no means an expert.
As a matter of fact "bandwidth" is ambiguous. There are many parameters that define bandwidth. 3 db, 6db, to name a few.
You see that an infinite sinusoid has zero bandwidth while a time limited sinusoid has finite bandwidth.
It feels as if you havent been through fourier analysis yet so I will try to explain it in different way.
Let us write Heisenberg's uncertainty principle in terms of frequency and time.
Δw*Δt ≥ 1/2
Now suppose that we have an infinite sinusoid running on entire time axis.
The frequency of this sinusoid can be determined very closely to the exact value (remember, the principle states that you can measure one of the conjugate variable accurately if other one is not accurate).
Now if we go on to decrease the length of this sinusoid we will be going away from the exact value of frequency. In extreme case let us assume we have a "sinusoid" such that it exists only for a quarter cycle and even less than that, (its not exactly an sinusoid, BTW) we see that we can never guess the exact frequency of sinusoid. Instead, a range of frequencies (that range too will increase if you just keep on cutting your sinusoid away). Thus a time limited signal has associated with it a range of frequencies.
I am sure this explanation is far from being perfect. I am open to criticisms.