Pure Mathematics F1: Limits

9 flashcards · Mathematics · by lakesare

1. Visually imagine punctured neighborhood Nr(c).

2. [Continuity] What's the relationship between: lim f(x), and continuity at c?

   approach to c = f(c)expectation = reality

3. [Sinc function] Visually imagine this limit.

4. Name 2 ways to define a limit.

   sequential definition ε-δ definition

5. Name 2 ways to define a function continuous at c.

   sequential definition ε-δ definition It is natural to ask which is the ‘better’ definition of continuity. It is difficult to give a definitive answer, but on the whole:• when proving the continuity of simpler functions the sequential definition is usually easier, whereas the ε-δ definition can work better with more complicated functions• when proving discontinuity the sequential definition is usually easier.

6. [Rules for finding limits] Visually imagine: combination rules for finding a limit composition rule for finding a limit You can find a limit from its definition (either sequential or ε-δ), or you can rely on the known limits and find a limit using various combination rules.

   composition rules - simple lim(f(x)*g(x)) combination rules - tricky lim(g(f(x)))

7. [Continuity at c, limit at c] What's the difference between: ε-δ definition of limit at c ε-δ definition of continuity at c

   Same, except limit: area around L, + ignore c itself continuity: area around f(c)

8. [Continuity at c, uniform continuity at c] Continuous VS uniformly continuous on I? f(x) is continuous at c

   The function is uniformly continuous on I when δ = ...ε... for any x=c on this interval. https://learn2.open.ac.uk/mod/forumng/discuss.php?d=3967786

9. [Limit at c] Visually imagine the ε-δ definition of a limit at c.