Differential Calculus: Limits, Continuity, Differentiability, Successive Differentiation and Expansion of Functions

Evaluating the limit as $x \to +\infty$:
Factor $x^2$ out of the square root in the denominator (for $x \gt 0$, $\sqrt{x^2} = x$):

Vertical asymptotes of a rational function occur at the values of $x$ where the denominator equals zero while the numerator is non-zero:

Since the numerator is a constant $-8 \ne 0$, both lines $x = 2$ and $x = -2$ ($x = \pm 2$) are vertical asymptotes.

According to Clairaut's (or Schwarz's) Theorem on mixed partial derivatives:
If a function $z = f(x, y)$ and its partial derivatives $f_x, f_y, f_{xy}, f_{yx}$ are continuous in an open region, then the mixed second-order partial derivatives are identical:

Since $z = 3x^2 y - x \sin xy$ is composed of polynomial and trigonometric functions, it is infinitely differentiable everywhere on $\mathbb{R}^2$.

Let $z = \sin u = \frac{x^2 y}{x + y}$.

Testing homogeneity of $z(x, y)$: