Asymptotic analysis

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Problem: Prove the equivalence of the formulas: \[ \int_{0}^{a} t^{\beta-1} e^{-\lambda t} d t \sim \Gamma(\beta) \lambda^{-\beta}, \quad \lambda \rightarrow+\infty, \quad a, \beta>0 . \]

2.12.1 Asymptotic analysis

3 $

Problem: Let function \( f(x) \in C(0 ; 1] \) be such that \[ f(x) \sim x^{a-1}(x \rightarrow+0), 0

2.12.2 Asymptotic analysis

6.24 $

Problem: Write down the asymptotic formula for the function: \[ f(x)=\left(\frac{e^{x}-1}{x}\right)^{\frac{1}{x}}, \quad x \rightarrow 0 . \]

2.12.3 Asymptotic analysis

3.75 $

Problem: Write down the asymptotic formula for the function: \[ \begin{array}{l} f(x)=\sqrt{x^{4}+2 x^{2}-2 x}-x^{2} \cos \frac{x}{x^{2}+1}, \\ x \rightarrow+\infty . \end{array} \]

2.12.4 Asymptotic analysis

4.25 $

Problem: Find the asymptotics of the roots of the equation: \[ \cos x=\frac{1}{x}, \quad x>0 . \]

2.12.5 Asymptotic analysis

7.49 $

Problem: Write down the asymptotic representation of the function: \[ F(x)=\int_{x}^{+\infty} \frac{e^{-t}}{\sqrt{t^{3}+2}} d t, \quad x \rightarrow+\infty . \]

2.12.6 Asymptotic analysis

6.24 $

Problem: Write down the asymptotic representation of the function: \[ F(x)=\int_{1}^{x} \sqrt{t^{3}+t} d t, \quad x \rightarrow+\infty . \]

2.12.7 Asymptotic analysis

6.24 $

Problem: Compare the infinitesimals: \[ \alpha(x)=\ln \left(x^{2}+1\right), \quad \beta(x)=x^{3}-2 x, x \rightarrow 0 . \]

2.12.8 Asymptotic analysis

0 $

Problem: Compare the infinitesimals: \[ \alpha(x)=\sin \ln (x+1), \beta(x)=x^{2}+2 x, x \rightarrow 0 . \]

2.12.9 Asymptotic analysis

0.75 $

Problem: Compare the infinitesimals: \[ \begin{array}{l} \alpha(x)=\sqrt{\cos \ln x}-1, \\ \beta(x)=x^{3}-3 x+2, \quad x \rightarrow 1 . \end{array} \]

2.12.10 Asymptotic analysis

1.25 $

Problem: Select the main part of the form \( c x^{k} \) of the infinitesimal function: \[ \alpha(x)=\cos x-\cos 5 x, \quad x \rightarrow 0 . \]

2.12.14 Asymptotic analysis

1 $

Problem: Select the main part of the form \( c x^{k} \) of the infinitesimal function: \[ \alpha(x)=\sin \ln \left(x^{3}+1\right), \quad x \rightarrow 0 . \]

2.12.15 Asymptotic analysis

1.25 $

Problem: Select the main part of the form \( C(x+1)^{k} \) of the infinitesimal function: \[ \alpha(x)=\sqrt{\pi}-\sqrt{\cos ^{-1} x}, \quad x \rightarrow-1 . \]

2.12.16 Asymptotic analysis

1.75 $

Problem: Select the main part of the form \( C\left(\frac{1}{x}\right)^{k} \) of the infinitesimal function: \[ \alpha(x)=\tan \frac{\pi x^{3}+2 x-1}{3 x^{3}-x+2}-\sqrt{3}, \quad x \rightarrow+\infty . \]

2.12.17 Asymptotic analysis

1.5 $

Problem: Determine the order of smallness \( \alpha(x)=\sqrt{x^{2}+9}-3 \) with respect to \( \beta(x)=x \) when \( x \rightarrow 0 \).

2.12.11 Asymptotic analysis

1 $

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