Multivariable Calculus (Calc III) - Complete Semester Course - 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. #mathematics #multivariablecalculus #limits #limitsandcontinuity #limits_and_continuity #calculus #iitjammathematics #calculus3