Calculation of limits

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Problem: Compute the limits of the functions: 1. \( \lim _{x \rightarrow 2} \frac{2^{x}-x^{2}}{(x-2)^{2}} \), 2. \( \lim _{x \rightarrow 0} \frac{\tan ^{-1}(2 x)-\tan (2 x)}{x^{3}} \), 3. \( \lim _{x \rightarrow 0} \frac{x^{2}+2^{x}}{2 x^{3}-2} \).

2.1.1 Calculation of limits

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Problem: Compute the limit of the function \[ \lim _{x \rightarrow 0}\left(\frac{1}{x^{2}}-\frac{\cot x}{x}\right) \text {. } \]

2.1.2 Calculation of limits

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Problem: Find the limits of functions without using L'Hopital's rule. 1. \( \lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+1}+\sqrt{x}}{\sqrt[5]{x^{3}+x}-2 x} \), 2. \( \lim _{x \rightarrow 0} \frac{\cos 2 x-\cos 4 x}{3 x^{2}} \) 3. \( \lim _{x \rightarrow \infty}\left(\frac{2 x+5}{1+2 x}\right)^{5 x} \).

2.1.3 Calculation of limits

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Problem: Prove that the sequence \( \left\{x_{n}\right\} \) has no limit. \[ x_{n}=\frac{1+(-1)^{n}}{2}+\frac{\sin n}{\sqrt{n^{2}+1}} \text {. } \]

2.1.4 Calculation of limits

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Problem: Prove that the sequence \( \left\{x_{n}\right\} \) has no limit. \[ x_{n}=1+(-1)^{n}+\frac{1}{n^{2}} \text {. } \]

2.1.5 Calculation of limits

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Problem: Prove that the number \( a \) is not the limit of the sequence \( \left\{x_{n}\right\} \). \[ x_{n}=\frac{2^{n}+n}{4^{n}}, \quad a=1 . \]

2.1.6 Calculation of limits

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Problem: Prove that the number \( a \) is not the limit of the sequence \( \left\{x_{n}\right\} \). \[ x_{n}=\frac{3 n+1}{6^{n}+2}, \quad a=1 \text {. } \]

2.1.7 Calculation of limits

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Problem: 1. Based on the definition of the limit of a sequence, prove that the limit of the sequence \( \left\{x_{n}\right\}=\left\{\frac{7 \cdot 4^{n}+3}{5 \cdot 4^{n}+4}\right\} \) is equal to the number \( a=\frac{7}{5} \), or disprove this statement. 2. If the statement is true, then for \( \varepsilon=\frac{1}{10000} \) find an \( N \), such that for all \( n>N \) the inequality \( \left|x_{n}-a\right|<\varepsilon \) holds.

2.1.8 Calculation of limits

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Problem: Find the limit: \[ \lim _{n \rightarrow \infty} \frac{-n^{5}+n^{2}-4}{-5 n^{7}+8 n^{5}+7 n^{4}-n^{2}-3} . \]

2.1.9 Calculation of limits

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Problem: Find the limit: \[ \lim _{n \rightarrow \infty} \frac{(3 n+6)^{2}\left(3 n^{4}+6 n^{3}-6 n-5\right)}{\left(-2 n^{2}+8 n+3\right)(-3 n-7)^{4}} \]

2.1.10 Calculation of limits

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Problem: Find the limit: \[ \lim _{n \rightarrow \infty} \frac{2 \cdot 5^{-n}+4 \cdot 4^{-n}+2}{-4 \cdot 5^{-n}+5 \cdot 4^{-n}-4} . \]

2.1.11 Calculation of limits

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Problem: Find the limit: \[ \lim _{n \rightarrow \infty}\left(\frac{-4+9 n+n^{2}}{2 n-2}-\frac{7-4 n+6 n^{2}}{7 n+5}\right) \text {. } \]

2.1.12 Calculation of limits

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Problem: Find the limit: \[ \lim _{x \rightarrow 3} \frac{x^{2}-2 x-3}{x^{2}-x-6} \text {. } \]

2.1.13 Calculation of limits

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Problem: Find the limit: \[ \lim _{x \rightarrow \infty} \frac{6 x^{4}-5 x^{2}+6 x-3}{-7 x-2} \tan ^{3}\left(\frac{4}{3 x-4}\right) \text {. } \]

2.1.14 Calculation of limits

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Problem: Find the limit: \[ \lim _{n \rightarrow \infty}\left(\frac{-9 n+1}{-9 n-8}\right)^{6 n-5} \text {. } \]

2.1.15 Calculation of limits

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