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\noindent
\force\qquad\parbox{1.5in}{\bf Math 1110}
\hfill {\Large \bf Supplemental Exercises} \hfill
\parbox{1.5in}{\bf \hfill L'Hopital's Rule}\qquad\force

\vspace{0.3in}

\noindent
\force \qquad {\bf Directions:} Find the limit. Be careful! L'Hopital's rule is not always needed and may not (directly) apply!  

\vspace{0.15in}

\noindent
\force \hfill \parbox[c]{7in}{
{\it Note:} Clicking on \href{https://www.wolframalpha.com}{\ALPHA} will take you to Wolfram Alpha's solution. Clicking on \href{https://www.symbolab.com}{\SYMBO} will take you to Symbolab's solution. While Symbolab will show step-by-step solutions, it cannot solve all of these problems (at least at the time of the writing of this handout).
} \hfill \force

\begin{multicols}{3}
\begin{enumerate}
\item \myEx{\lim\limits_{x \to 3} \dfrac{x-3}{3x^2-13x+12}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+(x-3)%2F(3x%5E2-13x%2B12)+as+x+goes+to+3}{\ALPHA}\;\; % Anton 10.2 #4 answer 1/5
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto%203%7D%5Cleft(%5Cfrac%7Bx-3%7D%7B3x%5E%7B2%7D-13x%2B12%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to 0} \dfrac{2\cosh(x)-2}{1-\cos(2x)}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+(2cosh(x)-2)%2F(1-cos(2x))+as+x+goes+to+0}{\ALPHA}\;\;  % Anton 10.2 #14 answer 1/2
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto%200%7D%5Cleft(%5Cfrac%7B2cosh%5Cleft(x%5Cright)-2%7D%7B1-cos%5Cleft(2x%5Cright)%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to 0} \dfrac{x-\mathrm{arctan}(x)}{x^3}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+(x-arctan(x))%2Fx%5E3+as+x+goes+to+0}{\ALPHA}\;\;  % Anton 10.2 #20 answer 1/3
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto%200%7D%5Cleft(%5Cfrac%7Bx-arctan%5Cleft(x%5Cright)%7D%7Bx%5E%7B3%7D%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to 0} \dfrac{x+\tan(x)}{\sin(x)}}\;\; 
          \href{http://www.wolframalpha.com/input/?i=Limit%5B(x%2Btan(x))%2Fsin(x),+x+-%3E+0%5D}{\ALPHA}\;\;  % Stewart 4.4 #10 answer 2
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto%200%7D%5Cleft(%5Cfrac%7Bx%2Btan%5Cleft(x%5Cright)%7D%7Bsin%5Cleft(x%5Cright)%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{t \to 0} \dfrac{5^t-3^t}{t}}\;\; 
         \href{http://www.wolframalpha.com/input/?i=Limit%5B(5%5Ex-3%5Ex)%2Fx,+x+-%3E+0%5D}{\ALPHA}\;\;  % Stewart 4.4 #19 answer ln(5/3) 
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto%200%7D%5Cleft(%5Cfrac%7B5%5E%7Bx%7D-3%5E%7Bx%7D%7D%7Bx%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to 1} \dfrac{\ln(x)}{x-1}}\;\; 
          \href{http://www.wolframalpha.com/input/?i=Limit%5Bln(x)%2F(x-1),+x+-%3E+1%5D}{\ALPHA}\;\; % Stewart 4.4 Example 1 answer 1
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto%201%7D%5Cleft(%5Cfrac%7Bln%5Cleft(x%5Cright)%7D%7Bx-1%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to \infty} \dfrac{\ln(x)}{\sqrt[3]{x}}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit%5Bln(x)%2Fx%5E(1%2F3),+x+-%3E+infinity%5D}{\ALPHA}\;\; % Stewart 4.4 Example 3 answer 0
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto%20%5Cinfty%7D%5Cleft(%5Cfrac%7Bln%5Cleft(x%5Cright)%7D%7B%5Csqrt%5B3%5D%7Bx%7D%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{t \to 0} \dfrac{\sin(6x)}{\sinh(2x)}}\;\; 
         \href{http://www.wolframalpha.com/input/?i=Limit%5Bsin(x)%2Fsinh(x),+x+-%3E+0%5D}{\ALPHA}\;\; %  answer 3
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto0%7D%5Cleft(%5Cfrac%7Bsin%5Cleft(6x%5Cright)%7D%7Bsinh%5Cleft(2x%5Cright)%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to 0} \dfrac{e^x-1-x-x^2/2}{x^3}}\;\; 
         \href{http://www.wolframalpha.com/input/?i=Limit%5B(e%5Ex+-+1+-+x+-+x%5E2%2F2)%2Fx%5E3,+x+-%3E+0%5D}{\ALPHA}\;\; % Stewart 4.4 #22 answer 1/6
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto0%7D%5Cleft(%5Cfrac%7Be%5E%7Bx%7D-1-x-%5Cfrac%7Bx%5E%7B2%7D%7D%7B2%7D%7D%7Bx%5E%7B3%7D%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to 0} \dfrac{\mathrm{arcsin}(x)}{x}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit%5Barcsin(x)%2Fx,+x+-%3E+0%5D}{\ALPHA}\;\; % Stewart 4.4 #25 answer 1
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto0%7D%5Cleft(%5Cfrac%7Barcsin%5Cleft(x%5Cright)%7D%7Bx%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to 0} \dfrac{1-e^{2x}}{\sec(x)}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit%5B(1-e%5E(2x))%2Fsec(x),+x+-%3E+0%5D}{\ALPHA}\;\;  % Stewart 4.4 #36 answer 0
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto0%7D%5Cleft(%5Cfrac%7B1-e%5E%7B2x%7D%7D%7Bsec%5Cleft(x%5Cright)%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to 2} \dfrac{\ln(5x-9)}{x^3-8}}\;\; 
         \href{http://www.wolframalpha.com/input/?i=Limit+of+ln(5x-9)%2F(x%5E3-8)+as+x+goes+to+2}{\ALPHA}\;\; % Anton 10.2 #18 answer 5/12
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto2%7D%5Cleft(%5Cfrac%7Bln%5Cleft(5x-9%5Cright)%7D%7Bx%5E%7B3%7D-8%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to 1} \dfrac{\ln(x)}{\tan(\pi x)}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+ln(x)%2Ftan(pi+x)+as+x+goes+to+1}{\ALPHA}\;\; % Anton 10.2 #7 answer 1/pi
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto1%7D%5Cleft(%5Cfrac%7Bln%5Cleft(x%5Cright)%7D%7Btan%5Cleft(%5Cpi%20x%5Cright)%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to -\infty} x^2e^x}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+x%5E2*e%5Ex+as+x+goes+to+-infinity}{\ALPHA}\;\;  % Stewart 4.4 #38 answer 0
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto-%5Cinfty%7D%5Cleft(x%5E%7B2%7De%5E%7Bx%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to 0^+} \sin(x) \ln(x)}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+sin(x)ln(x)+as+x+goes+to+0%5E%2B}{\ALPHA}\;\;  % Stewart 4.4 #40 answer 0
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto0%5E%7B%2B%7D%7D%5Cleft(sin%5Cleft(x%5Cright)ln%5Cleft(x%5Cright)%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to \infty} x-\ln(x)}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+x-ln(x)+as+x+goes+to+infinity}{\ALPHA}\;\;  % Stewart 4.4 #49 answer infinity
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto%5Cinfty%7D%5Cleft(x-ln%5Cleft(x%5Cright)%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to \infty} x^{1/x}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+x%5E(1%2Fx)+as+x+goes+to+infinity}{\ALPHA}\;\;  % Stewart 4.4 #57 answer 1
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto%5Cinfty%7D%5Cleft(x%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to 0^+} x^{1/x}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+x%5E(1%2Fx)+as+x+goes+to+0%5E%2B}{\ALPHA}\;\;  % answer 0
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto0%5E%7B%2B%7D%7D%5Cleft(x%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to 0^+} (\cos(x))^{1/x^2}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+cos(x)%5E(1%2Fx%5E2)+as+x+goes+to+0%5E%2B}{\ALPHA}\;\; % Stewart 4.4 #61 answer 1/sqrt(e)
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto0%5E%7B%2B%7D%7D%5Cleft(cos%5Cleft(x%5Cright)%5E%7B%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%7D%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to 0} (1+x)^{1/x}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+(1%2Bx)%5E(1%2Fx)+as+x+goes+to+0}{\ALPHA}\;\; % answer e
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto0%7D%5Cleft(%5Cleft(1%2Bx%5Cright)%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to 0^+} x^{\sin(x)}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+x%5Esin(x)+as+x+goes+to+0%5E%2B}{\ALPHA}\;\; % Anton 10.3 #25 answer 1
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto0%5E%7B%2B%7D%7D%5Cleft(x%5E%7Bsin%5Cleft(x%5Cright)%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to (\pi/2)^-} (\tan(x))^{\cos(x)}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+(tan(x))%5Ecos(x)+as+x+goes+to+(pi%2F2)%5E-}{\ALPHA}\;\; % Anton 10.3 #29 answer 1
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%5E%7B-%7D%7D%5Cleft(tan%5Cleft(x%5Cright)%5E%7Bcos%5Cleft(x%5Cright)%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to \infty} (\ln(x))^{1/x}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+(ln(x))%5E(1%2Fx)+as+x+goes+to+infinity}{\ALPHA}\;\; % Anton 10.3 #30 answer 1
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto%5Cinfty%7D%5Cleft(%5Cleft(ln%5Cleft(x%5Cright)%5Cright)%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%5Cright)}{\SYMBO}
\item \myEx{\lim\limits_{x \to \infty} (3^x+5^x)^{1/x}}\;\;
         \href{http://www.wolframalpha.com/input/?i=Limit+of+(3%5Ex%2B5%5Ex)%5E(1%2Fx)+as+x+goes+to+infinity}{\ALPHA}\;\; % Anton 10.3 #32 answer 5
         \href{https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bx%5Cto%5Cinfty%7D%5Cleft(%5Cleft(3%5E%7Bx%7D%2B5%5E%7Bx%7D%5Cright)%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%5Cright)}{\SYMBO}
\end{enumerate}
\end{multicols}

\vfill

Answers:
\begin{multicols}{3}
\begin{enumerate}
\item $\lim\limits_{x \to 3} \dfrac{x-3}{3x^2-13x+12}=1/5$
\item $\lim\limits_{x \to 0} \dfrac{2\cosh(x)-2}{1-\cos(2x)}=1/2$
\item $\lim\limits_{x \to 0} \dfrac{x-\mathrm{arctan}(x)}{x^3}=1/3$
\item $\lim\limits_{x \to 0} \dfrac{x+\tan(x)}{\sin(x)}=2$ 
\item $\lim\limits_{t \to 0} \dfrac{5^t-3^t}{t}=\ln(5/3)$ 
\item $\lim\limits_{x \to 1} \dfrac{\ln(x)}{x-1}=1$ 
\item $\lim\limits_{x \to \infty} \dfrac{\ln(x)}{\sqrt[3]{x}}=0$
\item $\lim\limits_{t \to 0} \dfrac{\sin(6x)}{\sinh(2x)}=3$ 
\item $\lim\limits_{x \to 0} \dfrac{e^x-1-x-x^2/2}{x^3}=1/6$ 
\item $\lim\limits_{x \to 0} \dfrac{\mathrm{arcsin}(x)}{x}=1$
\item $\lim\limits_{x \to 0} \dfrac{1-e^{2x}}{\sec(x)}=0$
\item $\lim\limits_{x \to 2} \dfrac{\ln(5x-9)}{x^3-8}=5/12$ 
\item $\lim\limits_{x \to 1} \dfrac{\ln(x)}{\tan(\pi x)}=1/\pi$
\item $\lim\limits_{x \to -\infty} x^2e^x=0$
\item $\lim\limits_{x \to 0^+} \sin(x) \ln(x)=0$
\item $\lim\limits_{x \to \infty} x-\ln(x)=\infty$
\item $\lim\limits_{x \to \infty} x^{1/x}=1$
\item $\lim\limits_{x \to 0^+} x^{1/x}=0$
\item $\lim\limits_{x \to 0^+} (\cos(x))^{1/x^2}=1/\sqrt{e}$
\item $\lim\limits_{x \to 0} (1+x)^{1/x}=e$
\item $\lim\limits_{x \to 0^+} x^{\sin(x)}=1$
\item $\lim\limits_{x \to (\pi/2)^-} (\tan(x))^{\cos(x)}=1$
\item $\lim\limits_{x \to \infty} (\ln(x))^{1/x}=1$
\item $\lim\limits_{x \to \infty} (3^x+5^x)^{1/x}=5$
\end{enumerate}
\end{multicols}



\end{document}


\item $\lim\limits_{ \to }$
         \href{}{\ALPHA}\;\;

