Problem $2)34$ points, $25$ minutes

For a$-$d $(6$ points each$)$: Evaluate each limit using limit laws and theorems. You do not need to name the laws, unless you are justifying skipping steps. If a limit does not exist, justify why. If a limit is plus or minus infinity, state it$.$

a$.\underset{x-2}{lim}3x^{2}2x-1$

b$.\underset{x3}{lim}\frac{x^2-9}{x^2-5x-6}$

c$.\underset{x}{lim}\frac{\sqrt[\phantom{2}]{25x^{2}2x7}}{3x-1}$

d$.\underset{t0}{lim}\frac{4-\sqrt[\phantom{2}]{16t}}{t}$

e$.(10$ points$)$ Find all horizontal and vertical asymptotes of the function $f(x)=\frac{5x^2-15x}{x^2-2x-3}$

Problem $3)30$ points, $15$ minutes

a$.$ Prove that the following equation has a solution in the interval $0,2.$

$2^x-1=x^3-5x2$

b$.$ Show that the following function is continuous for all real numbers: explain why it is continuous for $x<mn>0</mn>$ and for $x>0,$ and use the definition of continuity to show that it is continuous for $x=0.$

$f(x)=\{\begin{array}{ccc}cosx& if& x<mn>0</mn>\\ 1& x^{3}if& x0\end{array}$