Question

find each function value and limit. use - ∞ or ∞ where appropriate. f(x) = (6x + 4)/(5x - 3) (a) f(10) (b) f(100) (c) lim(x→∞) f(x) (a) f(10) = (round to the nearest thousandth as needed.)

Explanation:

Step1: Substitute x = 10 into f(x)

$f(10)=\frac{6\times10 + 4}{5\times10-3}$
$=\frac{60 + 4}{50-3}=\frac{64}{47}\approx1.362$

Step2: Substitute x = 100 into f(x)

$f(100)=\frac{6\times100 + 4}{5\times100-3}$
$=\frac{600+4}{500 - 3}=\frac{604}{497}\approx1.215$

Step3: Find the limit as x approaches infinity

For $f(x)=\frac{6x + 4}{5x-3}$, divide both the numerator and denominator by x:
$\lim_{x
ightarrow\infty}f(x)=\lim_{x
ightarrow\infty}\frac{6+\frac{4}{x}}{5-\frac{3}{x}}$
As $x
ightarrow\infty$, $\frac{4}{x}
ightarrow0$ and $\frac{3}{x}
ightarrow0$. So $\lim_{x
ightarrow\infty}f(x)=\frac{6 + 0}{5-0}=\frac{6}{5}=1.2$

Answer:

(A) $f(10)\approx1.362$
(B) $f(100)\approx1.215$
(C) $\lim_{x
ightarrow\infty}f(x)=1.2$