Bài tập về giới hạn dãy số và hàm số

Bài 1.

a, \lim      \frac{\sqrt{4{{n}^{2}}+1}-2n-1}{\sqrt{{{n}^{2}}+4n+1}-n}

b, \lim      \frac{{{3}^{n}}-{{4}^{n}}}{{{3}^{n}}+{{4}^{n}}}

c, \lim \frac{{{(-2)}^{n}}+{{3}^{n}}}{{{(-2)}^{n+1}}+{{3}^{n+1}}}

d, \lim      (\sqrt{2}.\sqrt[4]{2}.\sqrt[8]{2}...\sqrt[n]{2})

e, \lim      \frac{\sqrt{2{{n}^{2}}+1}-\sqrt{{{n}^{2}}+1}}{n+1}.

Bài 2.

  1. \lim      (\sqrt{n+3}-\sqrt{n-1})
  2. \lim      (\sqrt{{{n}^{2}}+3}-n)
  3. \lim \left[      n(\sqrt{2{{n}^{2}}+3}-\sqrt{2{{n}^{2}}-1}) \right]
  4. \lim      \frac{1}{\sqrt{n}(\sqrt{n+1}-\sqrt{n-1})}
  5. \lim (\sqrt{{{n}^{2}}+5}-\sqrt{{{n}^{2}}+1}).

Bài 3.

15, \underset{x\to 0}\lim\frac{2{{x}^{3}}-4{{x}^{2}}+1}{{{x}^{2}}-3x-2}                     16.  \underset{x\to -1}\lim \frac{{{x}^{4}}-1}{{{x}^{5}}-x+1}                     1 7.  \underset{x\to 0}\lim\frac{\sqrt{1+{{x}^{2}}}-1}{x}

18.  \underset{x\to 2}\lim\frac{\sqrt{x+7}-3}{{{x}^{2}}-4}                    19.  \underset{x\to -1}\lim \frac{\sqrt{{{x}^{2}}+3}-2}{\sqrt[3]{x}+1}                    20.  \underset{x\to 8}\lim\frac{\sqrt{2x}-4}{\sqrt[3]{x}-2}

21.  \underset{x\to 0}\lim\frac{2\sqrt{1+x}-\sqrt[3]{8-x}}{x}       22.  \underset{x\to 0}\lim\frac{\sqrt{1+2x}-\sqrt[3]{1+3x}}{{{x}^{2}}}     23.  \underset{x\to 1}\lim\frac{\sqrt[3]{x-2}+\sqrt[3]{{{x}^{2}}-x+1}}{{{x}^{2}}-1}

24.  \underset{x\to 1}\lim\frac{\sqrt{5-{{x}^{3}}}-\sqrt[3]{{{x}^{2}}+7}}{{{x}^{2}}-1}            25.  \underset{x\to 2}\lim\frac{{{x}^{2}}-4x+4}{{{x}^{2}}-5x+6}             26.  \underset{x\to 1}\lim\frac{{{x}^{3}}-1}{{{x}^{2}}-2x+1}.

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