(Solved): 8. Determine whether each limit is a DETERMINATE or INDETERMINATE form, then evaluate the limit. If ...

8. Determine whether each limit is a DETERMINATE or INDETERMINATE form, then evaluate the limit. If appropriate, use L'Hospital's Rule. You must indicate why it is applicable, as well as indicate that you are using it. If L'Hospital's Rule is not applicable, use an appropriate algebraic or analytical method (a graph or a table of values is NOT sufficient justification). (a) ${\lim }_{x\to \infty }\frac{2x+1}{7-x^{3/2}}$ (b) ${\lim }_{x\to 1}\frac{\sin [2x-2]}{x-1}$ (c) ${\lim }_{x\to -\infty }(5x-1)e^{2x}$ (d) ${\lim }_{x\to 0^{+}}\frac{1+\ln x}{3-4x}$ (e) ${\lim }_{x\to 0}\frac{e^{4x}-1-4x}{x^2}$ (f) ${\lim }_{x\to \infty }{\left(2+\frac{1}{x}\right)}^x$ (g) ${\lim }_{x\to \infty }{\left(1+\frac{2}{x}\right)}^x$ 9. A cylindrical can without a top is made to contain $27{\mathrm{cm}}^3$ of liquid. Find the dimensions that will minimize the amount of metal required to make the can. Give an exact answer AND then give an approximate answer rounded to the nearest tenth of a centimeter.