Answer
(a) When we use the zoom function on a graphing calculator, we could estimate that $\lim\limits_{x \to \infty}(\sqrt{3x^2+8x+6}-\sqrt{3x^2+3x+1}) = 1.4$
(b) We could guess that $\lim\limits_{x \to \infty}(\sqrt{3x^2+8x+6}-\sqrt{3x^2+3x+1}) = 1.4434$
(c) $\lim\limits_{x \to \infty}(\sqrt{3x^2+8x+6}-\sqrt{3x^2+3x+1}) = \frac{5\sqrt{3}}{6}$
Work Step by Step
(a) When we use the zoom function on a graphing calculator, we could estimate that $\lim\limits_{x \to \infty}(\sqrt{3x^2+8x+6}-\sqrt{3x^2+3x+1}) = 1.4$
(b) We can evaluate $f(x)$ for increasing values of $x$:
$f(10) = \sqrt{3(10)^2+8(10)+6}-\sqrt{3(10)^2+3(10)+1} = 1.4535$
$f(100) = \sqrt{3(100)^2+8(100)+6}-\sqrt{3(100)^2+3(100)+1} = 1.4446$
$f(1000) = \sqrt{3(1000)^2+8(1000)+6}-\sqrt{3(1000)^2+3(1000)+1} = 1.4435$
$f(10,000) = \sqrt{3(10,000)^2+8(10,000)+6}-\sqrt{3(10,000)^2+3(10,000)+1} = 1.4434$
$f(100,000) =\sqrt{3(100,000)^2+8(100,000)+6}-\sqrt{3(100,000)^2+3(100,000)+1} = 1.4434$
$f(1,000,000) = \sqrt{3(1,000,000)^2+8(1,000,000)+6}-\sqrt{3(1,000,000)^2+3(1,000,000)+1} = 1.4434$
We could guess that $\lim\limits_{x \to \infty}(\sqrt{3x^2+8x+6}-\sqrt{3x^2+3x+1}) = 1.4434$
(c) We can calculate the value of the limit:
$\lim\limits_{x \to \infty}(\sqrt{3x^2+8x+6}-\sqrt{3x^2+3x+1})$
$= \lim\limits_{x \to \infty}(\sqrt{3x^2+8x+6}-\sqrt{3x^2+3x+1})\cdot \frac{(\sqrt{3x^2+8x+6}+\sqrt{3x^2+3x+1}}{(\sqrt{3x^2+8x+6}+\sqrt{3x^2+3x+1}}$
$= \lim\limits_{x \to \infty}\frac{(3x^2+8x+6)-(3x^2+3x+1)}{\sqrt{3x^2+8x+6}+\sqrt{3x^2+3x+1}}$
$= \lim\limits_{x \to \infty}\frac{5x+5}{\sqrt{3x^2+8x+6}+\sqrt{3x^2+3x+1}}$
$= \lim\limits_{x \to \infty}\frac{(5x+5)(\frac{1}{x})}{(\sqrt{3x^2+8x+6}+\sqrt{3x^2+3x+1})(\frac{1}{x})}$
$= \lim\limits_{x \to \infty}\frac{5+\frac{5}{x}}{(\sqrt{3x^2/x^2+8x/x^2+6/x^2}+\sqrt{3x^2/x^2+3x/x^2+1/x^2}}$
$= \lim\limits_{x \to \infty}\frac{5+\frac{5}{x}}{(\sqrt{3+8/x+6/x^2}+\sqrt{3+3/x+1/x^2}}$
$= \frac{5+0}{(\sqrt{3+0+0}+\sqrt{3+0+0}}$
$= \frac{5}{\sqrt{3}+\sqrt{3}}$
$= \frac{5}{2\sqrt{3}}$
$= \frac{5\sqrt{3}}{6}$
