In Exercises 8 through 17, determine the region of...

In Exercises 8 through 17, determine the region of continuity of $f$ and draw a sketch showing as a shaded region in $R^{2}$ the region of continuity of $f$.
f(x, y)=\frac{x}{\sqrt{4 x^{2}+9 y^{2}-36}}

Key Concepts

Inequalities Defining Regions in R²

Inequalities involving two variables define regions in the plane that can be open or closed depending on whether the inequality is strict. In this context, the condition derived from the radicand leads to an inequality that, when graphed, shows the region in R² where the function is properly defined and continuous.

Square Root Function and Radicand Constraints

Since the square root function is only defined for non-negative inputs in real numbers, any function that includes a square root must impose a condition on its radicand. When the square root appears in a denominator, an additional restriction is that the radicand must be strictly greater than zero to avoid division by zero.

Continuity in Multivariable Calculus

Continuity for functions of several variables requires that the function is defined at a point and that its limit as the variables approach that point equals the function’s value. For functions composed of continuous operations (such as addition, multiplication, and division, when the denominator is nonzero), continuity across the domain follows provided the points in the domain satisfy the necessary conditions.

Domain of a Multivariable Function

In the context of functions of several variables, the domain is the set of all points in R² for which the function is defined. For functions that involve operations like square roots and division, the domain is determined by the conditions required for these operations to be valid, such as ensuring the radicand is non-negative (and often strictly positive to avoid division by zero).

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