Problem Set - L'Hôpital's Rule

Problem Set - L'Hôpital's Rule

 State L'Hôpital's rule and give an example which shows how it is used. Explain why L'Hôpital's rule works. Hint: Expand the numerator and the denominator in terms of $\Delta x$. Give three examples which illustrate that a limit problem that looks like it is coming out to $0/0$ could really be getting closer and closer to almost anything and must be looked at in a different way. Give three examples which illustrate that a limit problem that looks like it is coming out to ${1}^{\infty }$ could really be getting closer and closer to almost anything and must be looked at in a different way. Give three examples which illustrate that a limit problem that looks like it is coming out to ${0}^{0}$ could really be getting closer and closer to almost anything and must be looked at in a different way. Evaluate $\underset{x\to 1}{\mathrm{lim}}\frac{{x}^{2}+3x-4}{x-1}$. Evaluate $\underset{x\to 1}{\mathrm{lim}}\frac{{x}^{a}-1}{{x}^{b}-1}$. Evaluate $\underset{x\to 1}{\mathrm{lim}}\frac{\mathrm{ln}x}{x-1}$. Evaluate $\underset{x\to \pi }{\mathrm{lim}}\frac{\mathrm{tan}x}{x-\pi }$. Evaluate $\underset{x\to 3\pi /2}{\mathrm{lim}}\frac{\mathrm{cos}x}{x-\left(3\pi /2\right)}$. Evaluate $\underset{x\to {0}^{+}}{\mathrm{lim}}\frac{\mathrm{ln}x}{\sqrt{x}}$. Evaluate $\underset{x\to \infty }{\mathrm{lim}}\frac{{\left(\mathrm{ln}x\right)}^{3}}{{x}^{2}}$. Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{{6}^{x}-{2}^{x}}{x}$. Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{{e}^{x}-1-x-\left({x}^{2}/2\right)}{{x}^{3}}$. Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}x-x}{{x}^{3}}$. Evaluate $\underset{x\to \infty }{\mathrm{lim}}\frac{\mathrm{ln}\left(1+{e}^{x}\right)}{5x}$. Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{tan}\alpha x}{x}$. Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{2x-\mathrm{arcsin}x}{2x-\mathrm{arccos}x}$. Evaluate . Evaluate . Evaluate $\underset{x\to \infty }{\mathrm{lim}}{x}^{3}{e}^{-{x}^{2}}$. Evaluate $\underset{x\to \infty }{\mathrm{lim}}\left(x-\pi \right)\mathrm{cot}x$. Evaluate $\underset{x\to 0}{\mathrm{lim}}{x}^{-4}-{x}^{-2}$. Evaluate $\underset{x\to 0}{\mathrm{lim}}{x}^{-1}-\mathrm{csc}x$. Evaluate $\underset{x\to \infty }{\mathrm{lim}}x-\sqrt{{x}^{2}-1}$. Evaluate $\underset{x\to \infty }{\mathrm{lim}}\left(\frac{{x}^{3}}{{x}^{2}-1}-\frac{{x}^{3}}{{x}^{2}+1}\right)$. Evaluate $\underset{x\to {0}^{+}}{\mathrm{lim}}{x}^{\mathrm{sin}x}$. Evaluate $\underset{x\to 0}{\mathrm{lim}}{\left(1-2x\right)}^{1/x}$. Evaluate $\underset{x\to \infty }{\mathrm{lim}}{\left(1+3/x+5/{x}^{2}\right)}^{x}$. Evaluate $\underset{x\to \infty }{\mathrm{lim}}{x}^{1/x}$. Evaluate $\underset{x\to {0}^{+}}{\mathrm{lim}}{\left(\mathrm{cot}x\right)}^{\mathrm{sin}x}$. Evaluate $\underset{x\to \infty }{\mathrm{lim}}{\left(\frac{x}{x-1}\right)}^{x}$. Evaluate $\underset{x\to {0}^{+}}{\mathrm{lim}}{\left(-\mathrm{ln}x\right)}^{x}$.