Question

solve the limit of the above question.​

Answer 1

Explanation :

$\rm \star \: \: lim_{x \to1} \bigg( \cfrac{ \sqrt{x + 3} - 2}{x {}^{3} - 1 } \bigg) \\ \\ \\ \rm \underline{Applying \: \: L-hospital \: \: rule, } \\ \\ \\ \longrightarrow \sf \: lim_{x \to1} \bigg( \cfrac{ \cfrac{d}{dx}( \sqrt{x + 3} - 2) }{ \dfrac{d}{dx} (x {}^{3} - 1) } \bigg) \\ \\ \\ \longrightarrow \sf \: lim_{x \to1} \bigg( \cfrac{ \cfrac{1}{2 \sqrt{x + 3} } }{3x {}^{2} } \bigg) \\ \\ \\ \longrightarrow \sf \: lim_{x \to1} \bigg( \cfrac{1}{6x {}^{2} \sqrt{x + 3} } \bigg) \\ \\ \\ \longrightarrow\sf \: \bigg( \cfrac{1}{6 \times 1 {}^{2} \times \sqrt{1 + 3} } \bigg) \\ \\ \\ \longrightarrow\sf \: \bigg( \cfrac{1}{6 \times \sqrt{4} } \bigg) \\ \\ \\ \longrightarrow\sf \: \bigg( \cfrac{1}{6 \times 2} \bigg) \\ \\ \\ \longrightarrow\sf \: \bigg( \cfrac{1}{12} \bigg) \: \star$

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$\boxed{\begin{array}{|c|} \underline{ \tt \red{Some \: important\: limits, }} \\ \\ \\ \bull \tt lim_{x \to 0}(sinx) = 0 \\ \\ \\ \bull \tt lim_{x \to 0}(cosx) = 1 \\ \\ \\ \bull \tt lim_{x \to 0} \dfrac{sinx}{x} = 1 = lim_{x \to0} \dfrac{x}{sinx} \\ \\ \\ \bull \tt lim_{x \to 0} \dfrac{tanx}{x} = 1 = lim_{x \to0 } \dfrac{x}{tanx} \\ \\ \\ \bull \tt lim_{x \to 0} \dfrac{log(1 + x)}{x} = 1 \\ \\ \\ \bull \tt lim_{x \to 0}e {}^{x} = 1 \\ \\ \\ \bull \tt lim_{x \to 0} \dfrac{e {}^{x} - 1 }{x} = 1 \\ \\ \\ \bull \tt lim_{x \to 0} \dfrac{a {}^{x} - 1}{x} = log_{e}(a) \\ \\ \\ \bull \tt lim_{x \to 0} \bigg(1 + \dfrac{1}{x} \bigg) {}^{x} = e \end{array}}$

Answer 2

$\Large \mathfrak{ \color{green}{Solution : }}$

$\begin{gathered}{ \boxed{\rm { \red{\: \: lim_{x \to1} \bigg( \cfrac{ \sqrt{x + 3} - 2}{x {}^{3} - 1 } \bigg)}}}} \\ \\ \\ \rm \underline{ \purple{{We \: have \: to \: apply \: \: L-hospital \: \: rule, }}} \\ \\ \\ { \blue{\implies \sf \: lim_{x \to1} \bigg( \cfrac{ \cfrac{d}{dx}( \sqrt{x + 3} - 2) }{ \dfrac{d}{dx} (x {}^{3} - 1) } \bigg)}} \\ \\ \\{ \blue{ \implies\sf \: lim_{x \to1} \bigg( \cfrac{ \cfrac{1}{2 \sqrt{x + 3} } }{3x {}^{2} } \bigg) }}\\ \\ \\{ \blue{ \implies\sf \: lim_{x \to1} \bigg( \cfrac{1}{6x {}^{2} \sqrt{x + 3} } \bigg)}} \\ \\ \\ { \blue{\implies\sf \: \bigg( \cfrac{1}{6 \times 1 {}^{2} \times \sqrt{1 + 3} } \bigg)}} \\ \\ \\ { \blue{\implies\sf \: \bigg( \cfrac{1}{6 \times \sqrt{4} } \bigg)}} \\ \\ \\{ \blue{ \implies\sf \: \bigg( \cfrac{1}{6 \times 2} \bigg) }}\\ \\ \\ { \red{\implies\sf \: \bigg( \cfrac{1}{12} \bigg)}} \: \end{gathered}$