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[Гост]

Бихте ли ми помогнали с тези граници ?

[tex]\lim_{x\to\0}(cosx)^{\frac{1}{x^{2} } }[/tex]
[tex]\lim_{x\to\infty }\sqrt[3]{x^{3}-3x^{2}}-\sqrt{x^{2}-2x}[/tex]
[tex]\lim_{x\to\1}(1-x)\log_x 2[/tex]


Много ще съм Ви благодарна!!!

[Гост]

[tex]1. \quad \lim_{x \to 0} (\cos x)^{\frac{1}{x^2}} \quad \left [ 1^{+\infty}\right ][/tex]

[tex]\lim_{x \to x_{0}} \left [ f(x) \right ] ^ {g(x)} \quad \left [ 1^{+\infty} \right ] \Rightarrow \lim_{x \to x_{0}} \left [ f(x) \right ] ^ {g(x)} = e^{a}, \quad a = \lim_{x \to x_{0}} g(x) \left [ f(x) - 1 \right ][/tex]

[tex]a = \lim_{x \to x_{0}} \frac{1}{x^2} \left ( \cos x - 1 \right ) = - \lim_{x \to x_{0}} \frac{1}{x^2} \cdot 2 \sin^2 \frac{x}{2} = -\frac{1}{2}[/tex]

[tex]e^{a} = \frac{1}{\sqrt{e}}[/tex]

[tex]2. \quad \lim_{x \to 1} (1-x) \log_{x}2 = \lim_{x \to 1} (1-x) \frac{\ln 2}{\ln x} \sim \lim_{x \to 1} \frac{-\ln 2}{\frac{1}{x}} = -\lim_{x \to 1} x \ln 2 = -\ln 2[/tex]