Mathematics

Problem

For the function

h(x)=\frac{2x^{2}+1}{x^{2}+3},

determine a concrete numerical bound for

x

that guarantees the function value is within

0.001

of its limiting value at infinity. a. (i) Find the smallest real number

M

(to the nearest integer) such that

\left|\frac{2x^{2}+1}{x^{2}+3}-2\right|<0.001

whenever

x>M

. Show all steps leading to your answer. b. (ii) Verify explicitly that the inequality in part (i) indeed holds for

x=M