<p>You use L’hospital’s rule. For number one you don’t need to rewrite it before applying L’hospital. After L’hospital’s rule, you have -.5(4+x)^-.5, which is -1/4 after you substitute x=0.</p>
<p>For number two you have to rewrite it so it’s in an indeterminate form; I suggest writing it as sin(1/x)/(1/x). After L’hospital, you have -cos(1/x)(1/x^2)/(-1/x^2), and after substituting x=inf you get 1.</p>