Math, asked by whitepearl434, 5 months ago

solve the limit of the above question.​

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Answered by Anonymous
8

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}}

Answered by anshu24497
3

 \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}

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