L'Hopital's Rule and Related Rates of Growth

Find the limit of  $\lim_{x\rightarrow0}\left(\frac{\sin5x}{\sin2x}\right).$  

 Find the limit of $\lim_{x\rightarrow\infty}\left(\frac{\ln x}{x}\right).$ 

 Find the limit of $\lim_{x\rightarrow\infty}\left(\frac{2x^3+5x^2-7}{x^2-8x+5}\right).$ 

Find the limit:

Determine the convergence or divergence of the sequence. If it converges, find its limit.

 $a_n=\frac{3n^2-1}{2n^2+1}$  

Determine whether f grows faster than, slower than, or at the same rate as g as $x\rightarrow\infty.$  
 $f\left(x\right)=10x^3+2x^2,\ g\left(x\right)=e^x$  

Determine whether f grows faster than, slower than, or at the same rate as g as $x\rightarrow\infty.$  
 $f\left(x\right)=x,\ g\left(x\right)=5x$  

Determine whether f grows faster than, slower than, or at the same rate as g as $x\rightarrow\infty.$  
 $f\left(x\right)=x,\ g\left(x\right)=\tan^{-1}\left(x\right)$  

 $\frac{d}{dx}\left(\frac{1}{2x}\right)=$  

(a)  

 $\int_{ }^{ }e^{5x}dx=$  

(a)