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Evaluate    \mathrm{\lim _{t \rightarrow 1} \frac{\sqrt{2 t^2-1} \sqrt[3]{4 t^3-3 t}-1}{t^2-1}}

Option: 1

0


Option: 2

\frac{5}{2}


Option: 3

\frac{2}{3}


Option: 4

1


Answers (1)

best_answer

\mathrm{\lim _{t \rightarrow 1} \frac{\sqrt{2 t^2-1} \times \sqrt[3]{4 t^3-3 t}-1}{t^2-1} \\ }

\mathrm{ =\lim _{t \rightarrow 1} \frac{\sqrt[3]{4 t^3-3 t}-\frac{1}{\sqrt{2 t^2-1}}}{\frac{t^2-1}{\sqrt{2 t^2-1}}} }

\mathrm{ =\lim _{t \rightarrow 1} \frac{4 t\left(t^2-1\right)+t-\frac{1}{\left(2 t^2-1\right) \times \sqrt{2 t^2-1}}}{\left(t^2-1\right) \times\left[\frac{\sqrt[3]{\left(4 t^3-3 t\right)^2}}{\sqrt{2 t^2-1}}+\frac{\sqrt[3]{4 t^3-3 t}}{2 t^2-1}+\frac{1}{\left(2 t^2-1\right) \times \sqrt{2 t^2-1}}\right]} \\ }
\mathrm{ =\lim _{t \rightarrow 1} \frac{4 t+\frac{t\left(2 t^2-1\right) \times \sqrt{2 t^2-1}-1}{\left(2 t^2-1\right) \times \sqrt{2 t^2-1} \times\left(t^2-1\right)}}{\left[\frac{\sqrt[3]{\left(4 t^3-3 t\right)^2}}{\sqrt{2 t^2-1}}+\frac{\sqrt[3]{4 t^3-3 t}}{2 t^2-1}+\frac{1}{\left(2 t^2-1\right) \times \sqrt{2 t^2-1}}\right]} }
\mathrm{ =\lim _{t \rightarrow 1} \frac{4 t}{\left[\frac{\sqrt[3]{\left(4 t^3-3 t\right)^2}}{\sqrt{2 t^2-1}}+\frac{\sqrt[3]{4 t^3-3 t}}{2 t^2-1}+\frac{1}{\left(2 t^2-1\right) \times \sqrt{2 t^2-1}}\right]} \\ }
\mathrm{ +\lim _{t \rightarrow 1} \frac{\frac{t\left(2 t^2-1\right) \times \sqrt{2 t^2-1}-1}{\left(2 t^2-1\right) \times \sqrt{2 t^2-1} \times\left(t^2-1\right)}}{\left[\frac{\sqrt{\left(4 t^3-3 t\right)^2}}{\sqrt{2 t^2-1}}+\frac{\sqrt[3]{4 t^3-3 t}}{2 t^2-1}+\frac{1}{\left(2 t^2-1\right) \times \sqrt{2 t^2-1}}\right]} }

\mathrm{ =\frac{4}{3}+\frac{1}{3} \lim _{t \rightarrow 1} \frac{t\left(2 t^2-2\right) \times \sqrt{2 t^2-1}+t \times \sqrt{2 t^2-1}-1}{\left(t^2-1\right) \times\left(2 t^2-1\right) \times \sqrt{2 t^2-1}} \\ }

\mathrm{ =\frac{4}{3}+\frac{1}{3} \lim _{t \rightarrow 1} \frac{2 t}{2 t^2-1}+\frac{1}{3} \lim _{t \rightarrow 1} \frac{t \times \sqrt{2 t^2-1}-1}{\left(t^2-1\right)\left(2 t^2-1\right) \sqrt{2 t^2-1}} }

\mathrm{ =\frac{4}{3}+\frac{2}{3}+\frac{1}{3} \lim _{t \rightarrow 1} \frac{2 t^4-t^2-1}{\left(t^2-1\right)\left(2 t^2-1\right) \sqrt{2 t^2-1} \times\left(\sqrt{2 t^4-t^2}+1\right)} }
\mathrm{ =2+\frac{1}{3} \lim _{t \rightarrow 1} \frac{\left(t^2-1\right)\left(2 t^2+1\right)}{\left(t^2-1\right)\left(2 t^2-1\right) \sqrt{2 t^2-1} \times\left(\sqrt{2 t^4-t^2}+1\right)} }
\mathrm{ =2+\frac{1}{3} \lim _{t \rightarrow 1} \frac{\left(2 t^2+1\right)}{\left(2 t^2-1\right) \sqrt{2 t^2-1} \times\left(\sqrt{2 t^4-t^2}+1\right)} \\ }
\mathrm{ =2+\frac{1}{3} \times \frac{3}{2}=2+\frac{1}{2}=\frac{5}{2} . }

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Nehul

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