1. Prove that under the transformation of independent variable \( x= \) \( \varphi(t) \), an \( n^{\text {th }} \)-order linear differential equation remains an \( n^{\text {th }} \)-order linear differential equation, and a homogeneous linear differential equation remains a homogeneous linear differential equation. Here \( x=\varphi(t) \) has continuous derivatives up to order \( n \) and \( \varphi^{\prime}(t) \neq 0 \).

53) \( \lim _{x \rightarrow \frac{\pi}{3}} \frac{2 \cos x-1}{\sqrt{3}-\tan x} \)

1. Find the fourior series of the fanction \( f(x)=\left\{\begin{array}{ll}-1 ; & -1<x<0 \\ 1 ; & 0<x<1\end{array}\right. \)

If A is differentiable vector function, then \( \nabla \times \nabla \times A \) Select one: \( \nabla \nabla \cdot A-\nabla^{2} A \) 0 \( \nabla^{2} A \) \( \nabla \nabla \cdot A \)

If a fo \( f(x) \) sefinet beloe is contincious at \( x=0 \) Find the volue of \( k \). \( f(x)=\left\{\begin{array}{l}\frac{1-\cos 2 x}{2 x^{2}}, x<0 \\ k, x=0 \\ \frac{x}{|x|}, x>0 .\end{array}\right. \)

If \( \phi \) and \( \psi \) are differentiable scalar functions of position \( x, y, z \) then \( \operatorname{grad} \phi+\psi \) is equal to Select one: \( \operatorname{div} \phi+\psi \) \( \operatorname{curl} \phi+\psi \) \( \operatorname{grad} \phi+\operatorname{grad} \psi \) \( \operatorname{div} \phi+\operatorname{div} \psi \)