Замечательные пределы: [latex] \lim_{n \to 0} \frac{sin \frac{x}{2} }{sin3x} \lim_{n \to 0} \frac{1-cos2x}{x^2} \lim_{n \to 0} \frac{sin^{2}x }{sin2x} \lim_{n \to 0} (\frac{2x+1}{2x-3})^{x} [/latex]

[latex] \lim_{x \to 0} \frac{\sin \frac{x}{2} }{\sin 3x} = \lim_{x \to 0} \frac{3x\cdot \frac{x}{2} \cdot\sin \frac{x}{2} }{3x\cdot \frac{x}{2} \cdot\sin3x} = \lim_{x \to 0} \frac{ \frac{x}{2} }{3x} = \frac{1}{6} [/latex] [latex] \lim_{x \to 0} \frac{1-\cos2x}{x^2} = \frac{1-1+2\sin^2x}{x^2}= \lim_{x \to 0} \frac{2\sin^2x}{x^2} =2 [/latex] [latex] \lim_{x \to 0} \frac{\sin^2x}{\sin2x}= \lim_{x \to 0} \frac{\sin^2x}{2\sin x\cos x} = \lim_{x \to 0} \frac{\sin x}{2\cos x}= \lim_{x \to 0} \frac{x\sin x}{2x\cos x} =0[/latex] [latex] \lim_{x \to 0} ( \frac{2x+1}{2x-3} )^x= \lim_{x \to 0} ( \frac{2x-3+4}{2x-3} )^x= \lim_{x \to 0} (1+ \frac{4}{2x-3} )^x=\\ = \lim_{x \to 0} (1+ \frac{4}{2x-3} )^{x\cdot \frac{4}{2x-3} \cdot \frac{2x-3}{4} }=e^{ \lim_{x \to 0} \frac{4x}{2x-3} }=e^0=1[/latex]