Calculus Volume 3 · 免费预览
极限与连续性
二元函数的极限要求无论沿何种趋近路径，函数值都稳定于同一个值；当不同路径给出不同结果时，极限不存在。连续性则要求极限等于函数值，单变量情形中的各个定理也可相应推广。
1. Two-variable limits
Two-variable limits
2. Continuity
Continuity
3. Techniques for existence
Techniques for existence
4. Continuity's working consequences
Continuity's working consequences
5. Path test purpose
Evaluating a two-variable limit along different paths, like y = mx, serves what purpose?
6. Path-dependent example
Along the line y = mx, the expression xy/(x^2 + y^2) tends to which value?
7. Continuity of polynomials
Polynomial functions of two variables are continuous where?
8. Polar coordinate technique
Converting to polar coordinates, x = r cos theta and y = r sin theta, helps evaluate limits at the origin when the expression is bounded by what?
9. Path test limitation
Why can the path test never prove a two-variable limit exists?
10. Polar radius name
In the polar substitution, r measures the distance from the origin. As a single letter, what is it? Answer with one lowercase letter.
11. Squeeze two-variable form
The squeeze theorem for two variables needs an upper bound and a lower bound both tending to what? Answer with one word.
12. Composition continuity
Continuous functions of continuous functions are continuous. As a single word, what theorem? Answer with one word.
13. Bounded times vanishing
If f(x, y) is bounded and g(x, y) tends to zero at the origin, the product f times g tends to what?
14. Essential discontinuity cause
A discontinuity where the limit fails to exist because different paths disagree is called essential. As two words, what kind? Answer like answer key.