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\begin{document}

   \begin{center}
     {\bf MTH112 -- TEST 2}
   \end{center}

\begin{flushright}
   Name: \rule[0in]{1.5in}{.01in}
\end{flushright} \vs

\noin HONOR CODE: On my honor, I have neither given nor received 
any aid on this examination. \\
Signature: \rule[0in]{1.5in}{.01in}

\noin Note: Show all work on exam in order to receive full credit. 

\begin{enumerate}

    \item  Find the limit. 

       \begin{enumerate}

             \item $\ds {\lim _{x\rightarrow \infty }\frac{e^{3x}-e^{-3x}}
                                                          {  e^{3x}+e^{-3x}}}$ \vspace{2cm}
                              
             \item $\ds {\lim _{x\rightarrow \infty} 
                            \left [\ln (2+x)-\ln (1+x)\right ]}$ \vspace{2cm}
 
             \item $\ds {\lim_{x\rightarrow 0^+ }
                        (\cot x)^{\sin x}}$ \vspace{2cm}

             \item $\ds {\lim_{x\rightarrow \infty}(x-\sqrt{x^2-1})}$\vspace{2cm}

       \end{enumerate}

\newpage
 
   
  \item Find the exact value of 
         
         \begin{enumerate}
              \item $\ds {\cos \;(2 \sin ^{-1}\frac{5}{13})}$\vspace{2cm}
             
              \item $\ds \sec \; (\arctan 2)$\vspace{2cm}

        \end{enumerate}

  \item  Let $f(x)= \ds {\frac{\ln x}{x}}$. (Note: You must formally do all steps.)

         \begin{enumerate}

                \item Find the domain of $f$. \vspace{2cm}
             

                \item Find the asymptotes of $f$. \vspace{2cm}

                \item Find the critical points of $f$, and the intervals on which the 
                      function is increasing, decreasing. \vspace{4cm}

                \item Find the inflection points of $f$, and the intervals on which the function 
                      is decreasing, increasing. \vspace{4cm}

                 \item Sketch the graph of $f$. \vspace{4cm}
    
         \end{enumerate}

 \item  Using Integration by Parts, evaluate the following integrals.

               \begin{enumerate}
    
                  \item $\ds {\int t^2 \ln t \; dt}$ \vspace{5cm}

                  \item $\ds {\int _{0}^{\pi /2} x\cos 2x \;dx}$ \vspace{5cm}

                  \item $\ds {\int \cos x\ln (\sin x) \; dx}$\vspace{5cm}

                  \item $\ds {\int \sin (\sqrt{x}) \;dx} $. (Hint: First make a substitution, 
                         then use integration by parts.)\vspace{5cm}

                \end{enumerate}

\item Evaluate the trigonometric integrals. 
    
               \begin{enumerate}

                  \item $\ds {\int _{0}^{\pi /4} \sin ^4x \cos ^2 x dx}$ \vspace{5cm}

                  \item $\ds {\int \tan ^2 x \; dx}$ \vspace{5cm}

                  \item $\ds {\int \tan x \sec ^3x \; dx}$ \vspace{5cm}   

                  \item $\ds {\int \sin 5x \sin 2x \; dx}$ \vspace{5cm} 

                  \item $\ds {\int _{0}^{\pi } \sin ^2 x \; dx}$ \vspace{4cm}    

              \end{enumerate}
 
 
\end{enumerate}
\end{document}




