Evaluate The Value of :
[tex]\displaystyle \sf{\lim_{x\ \to\ 0}\ \dfrac{x^2}{1\ -\ \sqrt[3]{1\ +\ x^2}}}[/tex]
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Evaluate The Value of :
[tex]\displaystyle \sf{\lim_{x\ \to\ 0}\ \dfrac{x^2}{1\ -\ \sqrt[3]{1\ +\ x^2}}}[/tex]
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GIVEN :–
[tex] \\ \implies \displaystyle \sf{\lim_{x\ \to\ 0}\ \dfrac{x^2}{1\ -\ \sqrt[3]{1\ +\ x^2}}} \\ [/tex]
TO FIND :–
• Value of limit = ?
SOLUTION :–
• Let the function –
[tex] \\ \sf \implies y = \displaystyle \sf{\lim_{x\ \to\ 0}\ \dfrac{x^2}{1\ -\ \sqrt[3]{1\ +\ x^2}}} \\ [/tex]
• Now rationalization of denominator –
• We know that –
[tex] \\ \sf \to {a}^{3} - {b}^{3} = (a - b)( {a}^{2} + ab + {b}^{2}) \\ [/tex]
[tex] \\ \sf \to (a - b) = \dfrac{{a}^{3} - {b}^{3}}{( {a}^{2} + ab + {b}^{2})} \\ [/tex]
• So that –
[tex] \\ \sf \implies y = \displaystyle \sf{\lim_{x\ \to\ 0}\ \dfrac{x^2[ {(1)}^{2} + (1)(\sqrt[3]{1\ +\ x^2}) + ({\sqrt[3]{1\ +\ x^2}})^{2} ]}{(1)^{3} \ -\ (\sqrt[3]{1\ +\ x^2})^{3} }} \\ [/tex]
[tex] \\ \sf \implies y = \displaystyle \sf{\lim_{x\ \to\ 0}\ \dfrac{x^2[1 + (\sqrt[3]{1\ +\ x^2}) + ({\sqrt[3]{1\ +\ x^2}})^{2} ]}{1\ -\ (1\ +\ x^2)}} \\ [/tex]
[tex] \\ \sf \implies y = \displaystyle \sf{\lim_{x\ \to\ 0}\ \dfrac{x^2[1 + (\sqrt[3]{1\ +\ x^2}) + ({\sqrt[3]{1\ +\ x^2}})^{2} ]}{1\ -\ 1\ - \ x^2}} \\ [/tex]
[tex] \\ \sf \implies y = \displaystyle \sf{\lim_{x\ \to\ 0}\ \dfrac{ \cancel{x^2}[1 + (\sqrt[3]{1\ +\ x^2}) + ({\sqrt[3]{1\ +\ x^2}})^{2} ]}{- \ \cancel{x^2}}} \\ [/tex]
[tex] \\ \sf \implies y = (-1) \displaystyle \sf{\lim_{x\ \to\ 0}[1 + (\sqrt[3]{1\ +\ x^2}) + ({\sqrt[3]{1\ +\ x^2}})^{2} ]} \\ [/tex]
• Now Apply limits –
[tex] \\ \sf \implies y = (-1)[1 + (\sqrt[3]{1\ +\ (0)^2}) + ({\sqrt[3]{1\ +\ (0)^2}})^{2}] \\ [/tex]
[tex] \\ \sf \implies y = (-1)[1 + (\sqrt[3]{1}) + ({\sqrt[3]{1}})^{2}] \\ [/tex]
[tex] \\ \sf \implies y = (-1) [1 + 1 + 1]\\ [/tex]
[tex] \\ \sf \implies y = (-1) [3]\\ [/tex]
[tex] \\ \large\implies { \boxed{ \sf y =-3}} \\ [/tex]
Answer :
[tex]-3[/tex]
Explanation :
We know, lim x => 0 ( 1 + x )^n ≈ ( 1 + nx )
=> [tex]\displaystyle \sf{\lim_{x\ \to\ 0}\ \dfrac{x^2}{1\ -\ \sqrt[3]{1\ +\ x^2}}}[/tex]
=> [tex]\displaystyle \sf{\lim_{x\ \to\ 0}\ \dfrac{x^2}{1\ -\ {1\ +\ x^2}}}1/3[/tex]
=> [tex]\displaystyle \sf{\lim_{x\ \to\ 0}\ \dfrac{x^2}{1\ -\ {1\ +\ 1/3 \ x^2}}}[/tex]
=> [tex]\displaystyle \sf{\lim_{x\ \to\ 0}\ \dfrac{x^2}{{1/3\ x^2}}}[/tex]
=> cancel ( [tex]x^2[/tex] )
=> [tex]-1 / 3[/tex]
=> [tex]-3[/tex]
☆ Refer the attachment ⬆️
☆ ❥ So, It's Done ❥ !!