Answer
\[4\]
Work Step by Step
\[\begin{gathered}
f\left( x \right) = 4 + \frac{3}{{{x^2} + 2}} \hfill \\
{\text{Evaluate }}f\left( x \right){\text{ for the given values and complete the table}}{\text{.}} \hfill \\
x = {10^0} \to f\left( {{{10}^0}} \right) = 4 + \frac{3}{{{{\left( {{{10}^0}} \right)}^2} + 2}} = 5 \hfill \\
x = {10^1} \to f\left( {{{10}^1}} \right) = 4 + \frac{3}{{{{\left( {{{10}^1}} \right)}^2} + 2}} \approx 4.0294 \hfill \\
x = {10^2} \to f\left( {{{10}^2}} \right) = 4 + \frac{3}{{{{\left( {{{10}^2}} \right)}^2} + 2}} \approx 4.00029 \hfill \\
x = {10^3} \to f\left( {{{10}^3}} \right) = 4 + \frac{3}{{{{\left( {{{10}^3}} \right)}^2} + 2}} \approx 4.000003 \hfill \\
x = {10^4} \to f\left( {{{10}^4}} \right) = 4 + \frac{3}{{{{\left( {{{10}^4}} \right)}^2} + 2}} \approx 4.00000003 \hfill \\
x = {10^5} \to f\left( {{{10}^5}} \right) = 4 + \frac{3}{{{{\left( {{{10}^5}} \right)}^2} + 2}} \approx 4 \hfill \\
x = {10^6} \to f\left( {{{10}^6}} \right) = 4 + \frac{3}{{{{\left( {{{10}^6}} \right)}^2} + 2}} \approx 4 \hfill \\
\boxed{\begin{array}{*{20}{c}}
x&{f\left( x \right)} \\
{{{10}^0}}&5 \\
{{{10}^1}}&{4.0294} \\
{{{10}^2}}&{4.00029} \\
{{{10}^3}}&{4.000003} \\
{{{10}^4}}&{4.00000003} \\
{{{10}^5}}&4 \\
{{{10}^6}}&4
\end{array}} \hfill \\
{\text{Therefore,}} \hfill \\
\mathop {\lim }\limits_{x \to \infty } f\left( x \right) = \mathop {\lim }\limits_{x \to \infty } \left( {4 + \frac{3}{{{x^2} + 2}}} \right) = 4 \hfill \\
{\text{Graph}} \hfill \\
\end{gathered} \]
