Problem 813 (difficulty: 3/10)

Calculate the following limits using L'Hospital's rule and also using the Taylor polynomial!

 (1)  \(\displaystyle \lim_{x\to 0} \frac{\sin x - x}{x^3}\),      (2)  \(\displaystyle \lim_{x\to0} \frac{\cos(x^2)-1}{x}\),      (3)  \(\displaystyle \lim_{x\to0} \frac{\cos (xe^x)-\cos (xe^{-x})}{x^3}\),      (4)  \(\displaystyle \lim_{x=\infty} \frac{1+\sqrt{x}+\sqrt[3]{x}}{1+\sqrt[3]{x}+\sqrt[4]{x}}\),      (5)  \(\displaystyle \lim_{x\to0} \frac{(1+x)^5-(1+5x)}{x^2+x^5}\),      (6)  \(\displaystyle \lim_{x\to0} \frac{\cos x - e^{-\frac{x^2}2} }{x^4}\),      (7)  \(\displaystyle \lim_{x\to0} \frac{e^x\sin x - x(1+x)}{x^3}\).

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