Question

(a) use a graph of

$f(x)=\sqrt{3 x^{2}+8 x+9}-\sqrt{3 x^{2}+3 x+5}$

to estimate the value of $\lim _{x \
ightarrow \infty} f(x)$ to one decimal place.

(b) use a table of values of $f(x)$ to estimate the limit to four decimal places.

(c) find the exact value of the limit.

Explanation:

Step1: Rationalize the function

Multiply and divide \(f(x)=\sqrt{3x^{2}+8x + 9}-\sqrt{3x^{2}+3x + 5}\) by \(\sqrt{3x^{2}+8x + 9}+\sqrt{3x^{2}+3x + 5}\)

$$
LATEXBLOCK0
$$

Step2: Divide numerator and denominator by \(x\)

$$
LATEXBLOCK1
$$

Step3: Find the limit as \(x

ightarrow\infty\)
As \(x
ightarrow\infty\), \(\frac{4}{x}
ightarrow0\), \(\frac{8}{x}
ightarrow0\), \(\frac{9}{x^{2}}
ightarrow0\), \(\frac{3}{x}
ightarrow0\) and \(\frac{5}{x^{2}}
ightarrow0\)

$$
LATEXBLOCK2
$$

Answer:

(a) \(1.4\)
(b) \(1.4434\)
(c) \(\frac{5}{2\sqrt{3}}\)