Question

the domain of f consists of the five real numbers 1, 2, 3, 4, and 5. the table defines the function f for these values. the function g is given by g(x) = 2 ln x.
part b
(i) find all values of x, as decimal approximations, for which g(x) = 3, or indicate that there are no such values.
(ii) determine the end behavior of g as x increases without bound. express your answer using the mathematical notation of a limit.

Explanation:

Step1: Solve \(g(x)=3\)

Given \(g(x) = 2\ln x\). Set \(2\ln x=3\). Then \(\ln x=\frac{3}{2}\). Using the property \(y = \ln x\Leftrightarrow x = e^{y}\), we have \(x = e^{\frac{3}{2}}\approx4.48\).

Step2: Analyze the end - behavior of \(y = g(x)\)

As \(x\to+\infty\), the function \(y = g(x)=2\ln x\). We know that the natural logarithm function \(y=\ln x\) has the property \(\lim_{x
ightarrow+\infty}\ln x=+\infty\). So, \(\lim_{x
ightarrow+\infty}g(x)=\lim_{x
ightarrow+\infty}2\ln x=+\infty\).

Answer:

(i) \(x\approx4.48\)
(ii) \(\lim_{x
ightarrow+\infty}g(x)=+\infty\)