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  • Solve lim<img alt="\mathrm{x \rightarrow \sqrt{2}\left(\frac{e^{x^2}+e^2\left(1-x^2\right)}{\left[\ln \left(x^2-3 \sqrt{2} x+5\right)\right]^2}\right)_{\mathbf{m}}}" src="https://entrancecorner.onc

Solve lim\mathrm{x \rightarrow \sqrt{2}\left(\frac{e^{x^2}+e^2\left(1-x^2\right)}{\left[\ln \left(x^2-3 \sqrt{2} x+5\right)\right]^2}\right)_{\mathbf{m}}}

Option: 1

\mathrm{2 e^2}


Option: 2

e^2


Option: 3

e^5


Option: 4

0


Answers (1)

best_answer

\mathrm{\lim _{x \rightarrow \sqrt{2}}\left(\frac{e^{x^2}+e^2\left(1-x^2\right)}{\left(\ln \left(x^2-3 \sqrt{2} x+5\right)\right)^2}\right)=\lim _{t \rightarrow 0}\left(\frac{e^{(t+\sqrt{2})^2}+e^2\left(1-(t+\sqrt{2})^2\right)}{\left(\ln \left((t+\sqrt{2})^2-3 \sqrt{2}(t+\sqrt{2})+5\right)\right)^2}\right)}
\mathrm{=\lim _{t \rightarrow 0}\left(\frac{e^{(t+\sqrt{2})^2}+e^2\left(-t^2-2 \sqrt{2} t-1\right)}{\ln ^2\left(t^2-\sqrt{2} t+1\right)}\right)}\mathrm{=\lim _{t \rightarrow 0}\left(\frac{e^2+2 \sqrt{2} e^2 t+5 e^2 t^2+o\left(t^2\right)+e^2\left(-t^2-2 \sqrt{ } 2 t-\right.}{2 t^2+o\left(t^2\right)}\right.}
\mathrm{=2 e^2}

Posted by

Irshad Anwar

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