Calculus Volume 3 · free preview
Limits and Continuity
Limits of functions of two variables require the value to settle regardless of the approach path, failing when different paths give different results, and continuity is the limit equaling the value, with the theorems of the single-variable case extending.
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.
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