Determine whether or not each of the following limits exist and find the values for those that do$.$ For each, prove your assertion $(i)$ using the definition of limit and $($ii$)$ using other results $($theorems$,$ etc.$)$ from Section $3\mathrm{.}1.$

$($a$)\underset{x}{lim}\frac{3x}{2\sqrt[\phantom{2}]{x^2+1}}$

$($b$)\underset{x}{lim}\frac{x^2}{x+1}$

$($c$)\underset{x1}{lim}\frac{2x^2-3}{3x+5}$

$($d$)\underset{x1}{lim}\frac{1-x}{1-\sqrt[\phantom{2}]{x}}$Determine whether or not each of the following limits exist and find the values for those that do$.$ For each, prove your assertion $(i)$ using the definition of limit and $($ii$)$ using other results $($theorems$,$ etc.$)$ from Section $3\mathrm{.}1.$

$($a$)\underset{x}{lim}\frac{3x}{2\sqrt[\phantom{2}]{x^2+1}}$

$($b$)\underset{x}{lim}\frac{x^2}{x+1}$

$($c$)\underset{x1}{lim}\frac{2x^2-3}{3x+5}$

$($d$)\underset{x1}{lim}\frac{1-x}{1-\sqrt[\phantom{2}]{x}}$Determine whether or not each of the following limits exist and find the values for those that do$.$ For each, prove your assertion $(i)$ using the definition of limit and $($ii$)$ using other results $($theorems$,$ etc.$)$ from Section $3\mathrm{.}1.$

$($a$)\underset{x}{lim}\frac{3x}{2\sqrt[\phantom{2}]{x^2+1}}$

$($b$)\underset{x}{lim}\frac{x^2}{x+1}$

$($c$)\underset{x1}{lim}\frac{2x^2-3}{3x+5}$

$($d$)\underset{x1}{lim}\frac{1-x}{1-\sqrt[\phantom{2}]{x}}$