The given limit is
[tex]\lim_{\theta ->0}\frac{cos(4 \theta)-1}{sin(7 \theta)}[/tex]
If we put 0 for theta , we will get 0/0 , which is not defined.
SO in this case we use L- Hospital's rule and differentiate both numerator and denominator. That is
[tex]\lim_{\theta ->0}\frac{(d/d \theta)(cos(4 \theta)-1)}{(d / d \theta)(sin(7 \theta))}[/tex]
[tex]\lim_{\theta ->0}\frac{-4sin(4 \theta)}{7 cos(7 \theta)}[/tex]
Using substitution property of limit
[tex]\frac{-4sin(4 *0)}{7 cos(7*0)}[/tex]
[tex]\frac{-4*0}{7 *1}=0[/tex]
Therefore the value of the limit is 0.
