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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 gradient of the function at the given point: \[ z=x^{2}+y^{2}, \quad M(3 ; 2) \text {. } \]

2.3.2 Gradient and directional derivative

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Problem: Calculate the gradient of function \( z=x y \cos x y \) at point \( M(0 ;-3) \).

2.3.19 Gradient and directional derivative

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Problem: Find the domain of the function: \[ y=\frac{\sqrt{x^{2}-16}}{\log _{2}(x+5)} . \]

2.7.1 Function properties

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Problem: Find the domain of function \( y=\sqrt{35-|3 x-11|} \).

2.7.2 Function properties

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Problem: Examine the convergence of the series. 1) \( \sum_{n=1}^{\infty} \frac{1}{2^{n}+1} \), 2) \( \sum_{n=2}^{\infty} \frac{2 n}{\sqrt{n^{2}-1}} \).

2.10.21 Number series

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Problem: Expand function \( f(x)=e^{2 x} \) into a power series in the vicinity of the point \( x_{0}=3 \).

2.8.20 Power series

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Problem: Do the points \( A(1 ; \Gamma ; 2), B(-1 ; H ; 3), C(0 ; \Gamma ; 4) \), \( D(2 ; 2 ; \mathrm{H}) \) lie on the same plane? \[ \Gamma=\mathrm{H}=2 \text {. } \]

3.3.1 Line and plane in space

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3 Problem: Calculate the integral: \[ \int_{-2 \pi}^{2 \pi} \sin ^{3} 2 x \sqrt{100-x^{2}} d x \]

9.3.3.1 Definite Integrals

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Problem: Complex numbers \( z_{1}, z_{2} \) are given. Find \[ z_{1}+z_{2}, z_{1}-z_{2}, z_{1} \cdot z_{2}, \frac{z_{1}}{z_{2}}, \] where \( z_{1}=2-3 i, z_{2}=5-4 i \).

10.3.1 Operations with complex numbers

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Problem: Determine the type of the equation at the point \( (1 ; 1) \). \[ x u_{x x}+2 x y u_{x y}+y u_{y y}=\cos u^{2} . \]

11.5.3.2 With variable coefficients

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Problem: Let \( (a, b)=1 \). Prove that \( (a+b, a-b)=1 \) or 2 .

12.2.8 Number theory

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Problem: Find the general solution of the linear homogeneous differential equation with constant coefficients: \[ y^{\prime \prime \prime}+5 y^{\prime \prime}-6 y^{\prime}=0 . \] clip2net.com

8.1.3.13 Higher order differential equations

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Problem: Find the general solution of the linear homogeneous system of differential equations: \[ \left\{\begin{array}{c} \dot{x}=-y \\ \dot{y}=x \end{array} .\right. \]

8.3.13 Systems of ordinary differential equations

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Problem: Prove the logical identity: \[ \overline{A \vee B}=\bar{A} \wedge \bar{B} \text {. } \]

6.3.5 Boolean algebra

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