Answer
$$\left( {\text{a}} \right)0,{\text{ }}\left( {\text{b}} \right)\frac{5}{4},{\text{ }}\left( {\text{c}} \right){\text{The limit does not exist}}$$
Work Step by Step
$$\eqalign{
& \left( {\text{a}} \right)\mathop {\lim }\limits_{x \to \infty } \frac{{5{x^{3/2}}}}{{4{x^2} + 1}} \cr
& {\text{Divide both the numerator and the denominator by }}{x^2} \cr
& \mathop {\lim }\limits_{x \to \infty } \frac{{5{x^{3/2}}}}{{4{x^2} + 1}} = \mathop {\lim }\limits_{x \to \infty } \frac{{\frac{{5{x^{3/2}}}}{{{x^2}}}}}{{\frac{{4{x^2}}}{{{x^2}}} + \frac{1}{{{x^2}}}}} = \mathop {\lim }\limits_{x \to \infty } \frac{{\frac{5}{{{x^{1/2}}}}}}{{4 + \frac{1}{{{x^{3/2}}}}}} \cr
& {\text{Evaluate the limit}} \cr
& \mathop {\lim }\limits_{x \to \infty } \frac{{\frac{5}{{{x^{1/2}}}}}}{{4 + \frac{1}{{{x^{3/2}}}}}} = \frac{0}{{4 + 0}} = 0 \cr
& \cr
& \left( {\text{b}} \right)\mathop {\lim }\limits_{x \to \infty } \frac{{5{x^{3/2}}}}{{4{x^{3/2}} + 1}} \cr
& {\text{Divide both the numerator and the denominator by }}{x^{3/2}} \cr
& \mathop {\lim }\limits_{x \to \infty } \frac{{5{x^{3/2}}}}{{4{x^{3/2}} + 1}} = \mathop {\lim }\limits_{x \to \infty } \frac{{\frac{{5{x^{3/2}}}}{{{x^{3/2}}}}}}{{\frac{{4{x^{3/2}}}}{{{x^{3/2}}}} + \frac{1}{{{x^{3/2}}}}}} = \mathop {\lim }\limits_{x \to \infty } \frac{5}{{4 + \frac{1}{{{x^{3/2}}}}}} \cr
& {\text{Evaluate the limit}} \cr
& \mathop {\lim }\limits_{x \to \infty } \frac{5}{{4 + \frac{1}{{{x^{3/2}}}}}} = \frac{5}{{4 + 0}} = \frac{5}{4} \cr
& \cr
& \left( {\text{c}} \right)\mathop {\lim }\limits_{x \to \infty } \frac{{5{x^{3/2}}}}{{4\sqrt x + 1}} \cr
& {\text{Divide both the numerator and the denominator by }}{x^{3/2}} \cr
& \mathop {\lim }\limits_{x \to \infty } \frac{{5{x^{3/2}}}}{{4\sqrt x + 1}} = \mathop {\lim }\limits_{x \to \infty } \frac{{\frac{{5{x^{3/2}}}}{{{x^{3/2}}}}}}{{\frac{{4\sqrt x }}{{{x^{3/2}}}} + \frac{1}{{{x^{3/2}}}}}} = \mathop {\lim }\limits_{x \to \infty } \frac{5}{{\frac{4}{x} + \frac{1}{{{x^{3/2}}}}}} \cr
& {\text{Evaluate the limit}} \cr
& \mathop {\lim }\limits_{x \to \infty } \frac{5}{{\frac{4}{x} + \frac{1}{{{x^{3/2}}}}}} = \frac{5}{{0 + 0}} = \infty \cr} $$
