What happens if we change the limits of integral in Fourier transform? By definition of Fourier transform

$$X(\omega)=\int_{-\infty}^\infty x(t) e^{-j\omega t} dt $$

Now what will happen to the answer of transform  for example in case of $x(t)= \cos(\omega_0 t)$  if limit is $0$ to $A$ instead of $-\infty$ to $\infty$? 

For $x(t)=\cos(\omega_0 t)$ its fourier transform is given by $ X(\omega)= \pi[\delta(\omega-\omega_0) + \delta(\omega+\omega_o)]$

so if the limit is changed will it effect the answer?
Yes, it will affect the answer. What you're suggesting is known as the short-time Fourier transform. In the sinusoidal case that you proposed, you will observe spectral leakage, as the truncation of the integral limits is equivalent to multiplication of the sinusoid by a rectangular window function. This multiplication in the time domain maps to convolution in the frequency domain. The Fourier transform of a rectangular window is a sinc function, so the convolution will yield two sinc functions centered at the locations of the impulses in your original answer.