Calculation of limits

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Problem: Find the limit of the function without using L'Hopital's rule: \[ \lim _{x \rightarrow 0} \frac{\sin ^{-1} 3 x}{5 x} \text {. } \]

2.1.16 Calculation of limits

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Problem: Find the limit of the function without using L'Hopital's rule: \[ \lim _{x \rightarrow \infty}\left(\frac{2 x-1}{2 x+1}\right)^{x} \text {. } \]

2.1.17 Calculation of limits

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Problem: Find the limit of the function using L'Hopital's rule: \[ \lim _{x \rightarrow 0} \frac{e^{x}+e^{-x}-2}{1-\cos x} \text {. } \]

2.1.18 Calculation of limits

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Problem: Find the limit: \[ \lim _{x \rightarrow 2} \frac{\log (5-2 x)}{\sqrt{10-3 x}-2} . \]

2.1.19 Calculation of limits

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Problem: Find the limit: \[ \lim _{x \rightarrow 1} \frac{\cos \frac{\pi x}{2}}{5-5^{x}} \]

2.1.20 Calculation of limits

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Problem: Give the definition of the one-sided limit in the language \( \varepsilon-\delta \).

2.1.21 Calculation of limits

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Problem: Find the limit: \[ \lim _{x \rightarrow-1} \frac{x^{3}-3 x-2}{x^{3}-2 x^{2}-x-2} \text {. } \]

2.1.22 Calculation of limits

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Problem: Find the limit: \[ \lim _{x \rightarrow \infty}\left(\sqrt[3]{(x+1)^{2}}-\sqrt[3]{(x-1)^{2}}\right) \]

2.1.23 Calculation of limits

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Problem: Find the limit: \[ \lim _{x \rightarrow \infty} \frac{\sin 7 \pi x}{\tan 8 \pi x} . \]

2.1.24 Calculation of limits

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Problem: Find the limit: \[ \lim _{x \rightarrow \alpha} \frac{\alpha^{x^{2}-\alpha^{2}}-1}{\tan \ln \frac{x}{\alpha}} \text {. } \]

2.1.25 Calculation of limits

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Problem: Find the limit: \[ \lim _{x \rightarrow 0} \sqrt[x^{2}]{2-\cos x} \text {. } \]

2.1.26 Calculation of limits

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Problem: Find the limits of the functions without using L'Hopital's rule. a) \( \lim _{x \rightarrow 3} \frac{x^{2}-9}{x^{2}+2 x-15} \), b) \( \lim _{x \rightarrow+\infty} \frac{\sqrt{x}+1}{\sqrt{x}} \), c) \( \lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+1}-\sqrt{x^{2}-1}\right) \), d) \( \lim _{x \rightarrow 0} \frac{5 x^{2}}{\arcsin 3 x} \), e) \( \lim _{x \rightarrow \infty}\left(\frac{x-8}{x}\right)^{\frac{x}{2}} \), f) \( \lim _{x \rightarrow a} \frac{x^{m}-a^{m}}{x^{n}-a^{n}} \).

2.1.27 Calculation of limits

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Problem: Find the limits of the function without using L'Hopital's rule. a) \( \lim _{x \rightarrow 1} \frac{2-x-x^{2}}{1-x} \), b) \( \lim _{x \rightarrow \infty} \frac{x^{3}+x^{2}+x+1}{2 x^{3}+4} \), c) \( \lim _{x \rightarrow 4} \frac{\sqrt{x}-2}{x-4} \), d) \( \lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x^{3}} \), e) \( \lim _{x \rightarrow 0}(1+2 x)^{\frac{3}{x}} \) f) \( \lim _{x \rightarrow 0} \frac{\ln x}{\cot x} \).

2.1.28 Calculation of limits

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