Question

lime (++)*2 10 lim (cosk) 3/2 xoo use IBP to integrate x Sin (5x) dx x sin (25x) dx

Answer 1

Let

$y=\lim_{x\rightarrow\infty}\left(1+\dfrac{1}x\right)^{x^2}$

Taking log on both sides, we have

$\log y= \lim_{x\rightarrow\infty}\log \left(1+\dfrac{1}x\right)^{x^2}$

0041 + 1) 501,= f 801 =

$\implies \log y= \lim_{x\rightarrow\infty} \dfrac{\log \left(1+\dfrac{1}x\right)}{1/x^2}\qquad (0/0 form)$

Apply L' Hopital's Rule, we have

(0-6) B log y = lim 1+ (1+1/2)(-2/3)

$\implies \log y= \lim_{x\rightarrow\infty} \dfrac{ \left(x^2\right)}{2(x+1)}$

Again apply L'Hopital's Rule, se have

$\implies \log y= \lim_{x\rightarrow\infty} \dfrac{ \left(2x\right)}{2}$

$\implies \log y= \infty$

$\implies y=e^{ \infty}=\infty$

Therefore, we have

$y=\lim_{x\rightarrow\infty}\left(1+\dfrac{1}x\right)^{x^2}=\infty$

10)

Let

$y=\lim_{x\rightarrow0}\left(\cos x\right)^{3/x^2}$

Taking log on both sides, we have

$\log y=\log \lim_{x\rightarrow0}\left(\cos x\right)^{3/x^2}$

$\implies \log y=\lim_{x\rightarrow0}\log \left(\cos x\right)^{3/x^2}$

$\implies \log y=\lim_{x\rightarrow0}(3/x^2)\log \left(\cos x\right)$

$\implies \log y=3\lim_{x\rightarrow0}\dfrac{\log \left(\cos x\right)}{x^2}$

$\implies \log y=3\lim_{x\rightarrow 0}\dfrac{\log \left(\cos x\right)}{x^2}\qquad (0/0 form)$

Apply L'Hopital's rule, we have

$\implies \log y=3\lim_{x\rightarrow 0}\dfrac{\sin x}{\cos x (2x)}$

$\implies \log y=(3/2)\lim_{x\rightarrow 0}\dfrac{\sin x}x\dfrac{1}{\cos x }$

$\implies \log y=(3/2)(1)\dfrac{1}{\cos 0 }=(3/2)(1)(1)=3/2$

$\implies y=e^{3/2}$

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