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

\vspace{0.3in}

Find the limit. Be careful! L'Hopital's rule is not always needed and may not (directly) apply!  

Clicking on the check marks will take you to Wolfram Alpha's solutions.

\begin{multicols}{3}
\begin{enumerate}
\item $\lim\limits_{x \to 3} \dfrac{x-3}{3x^2-13x+12}$
         \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
\item $\lim\limits_{x \to 0} \dfrac{2\cosh(x)-2}{1-\cos(2x)}$
         \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
\item $\lim\limits_{x \to 0} \dfrac{x-\mathrm{arctan}(x)}{x^3}$
         \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
\item $\lim\limits_{x \to 0} \dfrac{x+\tan(x)}{\sin(x)}$ 
          \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
\item $\lim\limits_{t \to 0} \dfrac{5^t-3^t}{t}$ 
         \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) 
\item $\lim\limits_{x \to 1} \dfrac{\ln(x)}{x-1}$ 
          \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
\item $\lim\limits_{x \to \infty} \dfrac{\ln(x)}{\sqrt[3]{x}}$
         \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
\item $\lim\limits_{t \to 0} \dfrac{\sin(6x)}{\sinh(2x)}$ 
         \href{http://www.wolframalpha.com/input/?i=Limit%5Bsin(x)%2Fsinh(x),+x+-%3E+0%5D}{\ALPHA} %  answer 3
\item $\lim\limits_{x \to 0} \dfrac{e^x-1-x-x^2/2}{x^3}$ 
         \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
\item $\lim\limits_{x \to 0} \dfrac{\mathrm{arcsin}(x)}{x}$
         \href{http://www.wolframalpha.com/input/?i=Limit%5Barcsin(x)%2Fx,+x+-%3E+0%5D}{\ALPHA} % Stewart 4.4 #25 answer 1
\item $\lim\limits_{x \to 0} \dfrac{1-e^{2x}}{\sec(x)}$
         \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
\item $\lim\limits_{x \to 2} \dfrac{\ln(5x-9)}{x^3-8}$ 
         \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
\item $\lim\limits_{x \to 1} \dfrac{\ln(x)}{\tan(\pi x)}$
         \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
\item $\lim\limits_{x \to -\infty} x^2e^x$
         \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
\item $\lim\limits_{x \to 0^+} \sin(x) \ln(x)$
         \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
\item $\lim\limits_{x \to \infty} x-\ln(x)$
         \href{http://www.wolframalpha.com/input/?i=Limit+of+x-ln(x)+as+x+goes+to+infinity}{\ALPHA}  % Stewart 4.4 #49 answer infinity
\item $\lim\limits_{x \to \infty} x^{1/x}$
         \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
\item $\lim\limits_{x \to 0^+} x^{1/x}$
         \href{http://www.wolframalpha.com/input/?i=Limit+of+x%5E(1%2Fx)+as+x+goes+to+0%5E%2B}{\ALPHA}  % answer 0
\item $\lim\limits_{x \to 0^+} (\cos(x))^{1/x^2}$
         \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)
\item $\lim\limits_{x \to 0} (1+x)^{1/x}$
         \href{http://www.wolframalpha.com/input/?i=Limit+of+(1%2Bx)%5E(1%2Fx)+as+x+goes+to+0}{\ALPHA} % answer e
\item $\lim\limits_{x \to 0^+} x^{\sin(x)}$
         \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
\item $\lim\limits_{x \to (\pi/2)^-} (\tan(x))^{\cos(x)}$
         \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
\item $\lim\limits_{x \to \infty} (\ln(x))^{1/x}$
         \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
\item $\lim\limits_{x \to \infty} (3^x+5^x)^{1/x}$
         \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
\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}

