Limits and continuity in multivariable calculus build on single-variable concepts, extending them to functions with multiple inputs. These ideas are crucial for understanding how functions behave as we approach specific points in higher-dimensional spaces.

Evaluating limits and determining continuity become more complex with multiple variables. We'll explore various techniques, from direct substitution to coordinate transformations, to tackle these challenges and gain insights into function behavior.

Understanding Limits and Continuity in Multivariable Calculus

Concept of multivariable limits

• Definition of a limit for functions of several variables
  • Describes behavior of function as input approaches specific point in multidimensional space
  • Formal epsilon-delta definition generalizes single-variable case to n-dimensional input
• Approaches to a point in multidimensional space
  • Function value may depend on path taken (linear, parabolic, spiral)
  • All possible approaches must yield same limit value for limit to exist
• Existence of limits
  • Limit exists when function approaches single value regardless of path
  • May not exist due to oscillation, different directional limits, or unbounded behavior
• Relationship between limits and function behavior
  • Examines function values in small region around point of interest
  • Reveals local trends and potential discontinuities

Evaluation of multivariable limits

• Direct substitution method
  • Works when function continuous at point of interest
  • Fails for indeterminate forms ($0/0$, $\infty/\infty$)
• Factoring and simplification
  • Common factor method removes shared terms in numerator and denominator
  • Rationalization technique eliminates radicals
• L'Hôpital's rule for multivariable functions
  • Applies to indeterminate forms
  • Uses partial derivatives with respect to each variable
• Polar coordinate transformation
  • Converts $(x,y)$ to $(r,\theta)$ form
  • Simplifies limits involving radial symmetry
• Squeeze theorem for multivariable functions
  • Bounds target function between two functions with known limits
  • Proves limit exists when bounding functions converge to same value

Continuity of multivariable functions

• Definition of continuity for multivariable functions
  • At a point requires function defined, limit exists, and limit equals function value
  • On a domain extends point-wise continuity to entire region
• Conditions for continuity
  • Function must be defined at point of interest
  • Limit must exist as point approached from all directions
  • Limit value must match function value at point
• Types of discontinuities
  • Removable discontinuity can be fixed by redefining function at single point
  • Jump discontinuity shows abrupt change in function value
  • Infinite discontinuity occurs when function approaches infinity
• Continuity of common multivariable functions
  • Polynomial functions always continuous on entire domain
  • Rational functions continuous except where denominator zero
  • Trigonometric functions generally continuous over their domains
• Continuity on different types of domains
  • Open sets exclude boundary points
  • Closed sets include all boundary points
  • Connected sets can be traversed without jumping between disconnected regions

Properties of continuous multivariable functions

• Algebraic operations on continuous functions
  • Sum and difference of continuous functions remain continuous
  • Product and quotient continuous except where denominator zero
• Composition of continuous functions
  • Resulting function continuous if both component functions continuous
  • Inner function must map to domain of outer function
• Intermediate Value Theorem for multivariable functions
  • Guarantees function takes on all values between two function values on connected domain
  • Used to prove existence of solutions to equations
• Extreme Value Theorem for multivariable functions
  • Ensures global maximum and minimum exist on closed, bounded sets
  • Critical in solving optimization problems
• Preservation of continuity under coordinate transformations
  • Continuity maintained when changing between coordinate systems (Cartesian to polar)
  • Allows flexibility in choosing most suitable coordinate system for problem