Question

it is a fact that the function (f(x)=\frac{5 + x^{2}}{5+4x^{2}}) has a limiting value. use a table of values to estimate the limiting value. (suggestion: we suggest starting the table at 0 and using a table increment of 1. round your answer to two decimal places.)

Explanation:

Step1: Consider limit as x approaches infinity

As \(x\to\pm\infty\), we analyze \(\lim_{x\to\pm\infty}\frac{5 + x^{2}}{5+4x^{2}}\). Divide numerator and denominator by \(x^{2}\), we get \(\lim_{x\to\pm\infty}\frac{\frac{5}{x^{2}}+ 1}{\frac{5}{x^{2}}+4}\).

Step2: Evaluate the limit of each term

As \(x\to\pm\infty\), \(\lim_{x\to\pm\infty}\frac{5}{x^{2}} = 0\). So \(\frac{\lim_{x\to\pm\infty}\frac{5}{x^{2}}+ 1}{\lim_{x\to\pm\infty}\frac{5}{x^{2}}+4}=\frac{0 + 1}{0+4}\).

Step3: Calculate the final result

\(\frac{0 + 1}{0+4}=\frac{1}{4}=0.25\)

Answer:

0.25