To evaluate the limit of the function \(\frac{x^2 - y^2}{x^2 + y^2}\) as \((x, y)\) approaches \((0, 0)\) along different paths, we will look at two specific paths:
When \(y = 0\), the function simplifies as follows: \[ \frac{x^2 - y^2}{x^2 + y^2} = \frac{x^2 - 0^2}{x^2 + 0^2} = \frac{x^2}{x^2} = 1 \]
So, along the path \(y = 0\), as \((x, y) \to (0, 0)\), the limit of the function is 1: \[ \lim_{(x, 0) \to (0, 0)} \frac{x^2 - 0^2}{x^2 + 0^2} = 1 \]
When \(x = 0\), the function simplifies as follows: \[ \frac{x^2 - y^2}{x^2 + y^2} = \frac{0^2 - y^2}{0^2 + y^2} = \frac{-y^2}{y^2} = -1 \]
So, along the path \(x = 0\), as \((x, y) \to (0, 0)\), the limit of the function is -1: \[ \lim_{(0, y) \to (0, 0)} \frac{0^2 - y^2}{0^2 + y^2} = -1 \]
The evaluation along these two paths yields different results:
I struggled initially with understanding path-dependent limits, but overcame this through research and reading.
The key takeaway from the course was understanding how diverse mathematical areas like probability, calculus, algebra, and linear regression converge to serve data science effectively. A particularly interesting example was the application of linear algebra in creating eigenimages for image compression. The final project really brought it all together, demonstrating the vital role of math in data science.