Function Limits, Sequence Convergence, and Their Role in Analysis

All flashcards

What is the criterion for continuity of a function at $c$?

The function is defined at $c$, and its limit there equals its value: $\lim_{x\to c}f(x)=f(c)$.

How is the derivative at $x$ defined using a limit?

$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$, when this limit exists.

How is an infinite series defined through partial sums?

Its sum is the finite limit of its sequence of partial sums, if that limit exists.

How do pointwise and uniform convergence of functions differ?

Pointwise convergence is checked separately at each input; uniform convergence controls approximation across the whole domain at once.

What does completeness mean for a metric space?

Every Cauchy sequence in the space converges to a point in that space.

Quiz questions

1. When does a two-sided limit at $c$ exist, according to the one-sided limit criterion?Answer: When both one-sided limits exist and are equal.
2. What quantity does the derivative definition measure as $h$ approaches zero?Answer: The instantaneous rate of change of the function.
3. What is guaranteed when continuous functions converge uniformly to a function?Answer: The limiting function is continuous.
4. What is true of a limit of a real-valued function at a point, if it exists?Answer: It is unique.
5. What does $\lim_{x\to+\infty}f(x)=L$ mean?Answer: For every $\varepsilon>0$, some $M$ ensures $|f(x)-L|<\varepsilon$ whenever $x>M$.