Question

derive the above limit​

Answer 1

☯ Explanation,

$\red\bigstar \tt \: lim_{x \rightarrow \: 1} \bigg( \dfrac{1}{1 - x} - \dfrac{3}{1 - x {}^{3} } \bigg) \\ \\ \\ : \implies \tt \: \: lim_{x \rightarrow \: 1} \bigg( \dfrac{1 + x + x {}^{2} - 3 }{(1 - x)(1 + x + x {}^{2} )} \bigg) \\ \\ \\ {\underline { \tt{Applying \: L - Hospital \: rule, }}} \\ \\ \\ : \implies \tt \: \: lim_{x \rightarrow \: 1} \bigg( \dfrac{ \dfrac{d}{dx} (1 + x + x {}^{2} - 3) }{ \dfrac{d}{dx} (1 - x)(1 + x + x {}^{2}) } \bigg) \\ \\ \\ : \implies \tt \: \: lim_{x \rightarrow \: 1} \bigg( \dfrac{1 + 2x}{ - 3x {}^{2} } \bigg) \\ \\ \\ : \implies \tt \: \bigg( \frac{1 + 2 \times 1}{-3 \times 1 {}^{2} } \bigg) \\ \\ \\ : \implies { \underline{ \tt{ \boxed{ - 1}}}} \: \green \bigstar$

☯ Hence,

$\dag \boxed{\tt \: lim_{x \rightarrow \: 1} \bigg( \dfrac{1}{1 - x} - \dfrac{3}{1 - x {}^{3}} \bigg) = -1}$

Answer 2

$\huge\mathcal\colorbox{orchid}{{\color{black}{★ANSWER★}}}$

$lim \: x→1 \: \: ( \frac{1}{1 - x} - \frac{3}{1 - x³} )$

$⇒lim \: x→1 \: \: ( \frac{1 + x + {x}^{2} - 3 }{(1 - x)(1 + x + {x}^{2} )} )$

Applying L- Hospital Rule,

$⇒lim \: x→1 \: \: ( \frac{ \frac{d}{dx} (1 + x + {x}^{2} - 3) }{ \frac{d}{dx}(1 - x)(1 + x + {x}^{2}) } )$

$⇒lim \: x→1 \: \: ( \frac{1 + 2x}{ - 3 {x}^{2} } )$

$⇒( \frac{1 + 2 \times 1}{ - 3 \times 1²} )$

$⇒ - 1$

Hence,

$lim \: x→1 \: \: ( \frac{1}{1 - x} - \frac{3}{1 - {x}^{3} } )= - 1$