$$\lim_{x \to 0}\left(\frac{\sin{x}-\ln({\text{e}^{x}}\cos{x})}{x\sin{x}}\right)$$
Can this limit be calculated without using L'Hopital's rule?
[Math] How to calculate $\lim_{x \to 0}\left(\frac{\sin{x}-\ln({\text{e}^{x}}\cos{x})}{x\sin{x}}\right)$ limit without using L’Hopital’s rule
analysiscalculuslimitslimits-without-lhopitalreal-analysis
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Best Answer
Hints: $\quad\ln(ab)=\ln a+\ln b,\qquad\ln e^x=x,\qquad\cos x=\sqrt{1-\sin^2x},\qquad\ln a^b=b\ln a,$
$\ln(1+t)\simeq t\quad$ when $\quad t\simeq0,\quad$ and $\quad\lim_{x\to0}\dfrac{\sin x}x=1$.