Question

Let $\lim_{x \rightarrow \infty}\frac{ax^{2} + bx + c}{dx^{2} + ex + f}$, then the values of $\frac{b}{e}$ and $\frac{c}{f}$ if $\frac{a}{d}$ is continuous at $\frac{d}{a}$ are respectively

• A. $\lim_{x \rightarrow \infty}\left\lbrack \sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x} \right\rbrack$
• B. $\frac{1}{2}$
• C. $\log 2$
• D. None of these

Answer 1

Answer: (B)

$\frac{1}{2}$

Answer 2

For $f ( x )$ to be continuous at $x = 0$

⇒ $\lim _ { x \rightarrow 0 ^ { - } } f ( x ) = f ( 0 ) = \lim _ { x \rightarrow 0 ^ { + } } f ( x )$

⇒

Now,

$\lim _ { x \rightarrow 0 ^ { + } } e ^ { \tan 2 x / \tan 3 x } = \lim _ { x \rightarrow 0 ^ { + } } e ^ { \left( \frac { \tan 2 x } { 2 x } - 2 x \right) / \left( \frac { \tan 3 x } { 3 x } 3 x \right) } = \lim _ { x \rightarrow 0 ^ { + } } e ^ { 2 / 3 } = e ^ { 2 / 3 }$

$\therefore$ $e ^ { a } = b = e ^ { 2 / 3 }$ ⇒ $a = \frac { 2 } { 3 }$ and $b = e ^ { 2 / 3 }$