Question

find lim x tends to27 (x^ 1/3 +3)(x^ 1/3 -3) /(x-27)​

Answer 1

Step-by-step explanation:

Identity (II)

$9 a^{2} - 24ab \: + 16b^{2}$

$(a - b) ^{2}$

$= a^{2} - 2ab + b ^{2}$

$\huge\fbox\pink{Answer}$

$(3a - 4b) (3a - 4b)$

Using identity (II)

$(3a) ^{2} - 2 \times 3a \times 4b + (4b) ^{2}$

$9a^{2} - 24ab + 16b ^{2}$

Answer 2

☯ Explanation,

$\tt \red \bigstar \: lim_{x \rightarrow \: 27} \bigg( \dfrac{(x {}^{ \dfrac{1}{3} } + 3)(x {}^{ \dfrac{1}{3} } - 3) } {(x - 27)} \bigg) \\ \\ \\{ \underline{ \red {\tt{Applying \: L - Hospital \: rule, }}}} \\ \\ \\ \mapsto \tt \: lim_{x \rightarrow \: 27} \bigg( \dfrac{ \dfrac{d}{dx} \bigg((x {}^{ \dfrac{1}{3} } + 3)(x {}^{ \dfrac{1}{3} } - 3) \bigg) } { \dfrac{d}{dx} \bigg(x - 27) \bigg)} \bigg) \\ \\ \\ \mapsto \tt \: lim_{x \rightarrow \: 27} \bigg( \dfrac{ \dfrac{2}{ 3\sqrt[3]{x} } }{1} \bigg) \\ \\ \\ \mapsto \tt \: lim_{x \rightarrow \: 27} \bigg( \dfrac{2}{3 \sqrt[3]{x} } \bigg) \\ \\ \\ \mapsto \tt \: \bigg( \dfrac{2}{3 \sqrt[3]{27} } \bigg) \\ \\ \\ \mapsto{ \boxed { \underline{ \tt{ \dfrac{2}{9} }}}} \: \green \bigstar$

☯ Hence,

$\dag { \boxed{ \tt{lim_{x \rightarrow \: 27} \bigg( \dfrac{(x {}^{ \dfrac{1}{3} } + 3)(x {}^{ \dfrac{1}{3} } - 3) } {(x - 27)} \bigg) = \dfrac{2}{9} }}}$