Question-46

Evaluate:

\[ \begin{equation*} \lim\limits _{x\rightarrow \infty } \ \sqrt{3x^{2} +8x+6} -\sqrt{3x^{2} +3x+1} \end{equation*} \]

• \(\cfrac{5}{2 \sqrt{3}}\)
• \(5\)
• \(\sqrt{3}\)
• \(\cfrac{1}{3}\)
• \(\cfrac{1}{\sqrt{3}}\)

Hint

Rationalise the numerator.

Answer

• \(\cfrac{5}{2 \sqrt{3}}\)
• \(5\)
• \(\sqrt{3}\)
• \(\cfrac{1}{3}\)
• \(\cfrac{1}{\sqrt{3}}\)

Solution

We have:

\[ \begin{equation*} \begin{array}{ l } \sqrt{3x^{2} +8x+6} -\sqrt{3x^{2} +3x+1}\\ \\\\ =\cfrac{5( x+1)}{\sqrt{3x^{2} +8x+6} +\sqrt{3x^{2} +3x+1}}\\ \\\\ =\cfrac{5\left( 1+\cfrac{1}{x}\right)}{\sqrt{3+\cfrac{8}{x} +\cfrac{6}{x^{2}}} +\sqrt{3+\cfrac{3}{x} +\cfrac{1}{x^{2}}}} \end{array} \end{equation*} \]

Now:

\[ \begin{equation*} \lim\limits _{x\rightarrow \infty } \ \cfrac{5\left( 1+\cfrac{1}{x}\right)}{\sqrt{3+\cfrac{8}{x} +\cfrac{6}{x^{2}}} +\sqrt{3+\cfrac{3}{x} +\cfrac{1}{x^{2}}}} =\cfrac{5}{2\sqrt{3}} \end{equation*} \]