Multivariable limits and continuity

Limits and continuity for scalar-valued functions of multiple variables, like z=f(x,y). Showing when the limit does and does not exist (including using polar coordinates). Continuity. Multivariable Calculus Unit 3 Lecture 5.

This lecture covers the concept of limits in multivariable calculus, crucial for understanding differentiation (our next topic). We explore how to compute limits for scalar and vector-valued functions, transitioning from the simpler case of single-variable calculus to the complexities of multiple variables. Through examples, we demonstrate techniques for establishing the existence or non-existence of limits in multivariable functions.

- Single Variable Calculus: The limit of a function as it approaches a point 𝑎 is analyzed from the left and right.
- Multivariable Calculus: The limit as 𝑥⃗ approaches a point 𝑎⃗ in ℝ𝑛 is more complex due to the multidimensional nature of the domain.

Examples of Limits in Multivariable Calculus

1. Function 𝑓(𝑥,𝑦)=𝑥^2−𝑦^2: Approaching along the x-axis and y-axis gives different limits, indicating the overall limit does not exist.

2. Function 𝑓(𝑥,𝑦)=𝑥^2𝑦/(𝑥^4+𝑦^2): Approaching along different paths yields zero, but this is not conclusive. Using polar coordinates, we show the limit is zero as 𝑟 approaches zero, regardless of 𝜃.

Establishing Non-Existence and Existence of Limits
- To show a limit does not exist, find two paths leading to different limits.
- To prove a limit exists (two-dimensional domains), use polar coordinates to show the limit as 𝑟 approaches zero is consistent, regardless of 𝜃.

Defining Continuity: A function 𝑓(𝑥⃗ ) is continuous at 𝑎⃗ if 𝑓(𝑎⃗ ) exists, the limit as 𝑥⃗ approaches 𝑎⃗ exists, and these two values are equal: lim(𝑥⃗ →𝑎⃗) 𝑓(𝑥⃗ )=𝑓(𝑎⃗ ). Familiar functions like polynomials, trigonometric, exponential, logarithmic, and rational functions are continuous on their domains.

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[sethsherman-ekpo]

Wish you were teaching thirty years ago, but better late than never! Thank you.