Limits and Continuity flashcards

Question 1

What does it mean that $\lim_{x	o a} f(x)=L$?

Accepted Answer

For every $\varepsilon>0$ there exists $\delta>0$ such that if $0<|x-a|<\delta$, then $|f(x)-L|<\varepsilon$.

Question 2

Compute $\lim_{x	o 2}\frac{x^2-4}{x-2}$.

Accepted Answer

$4$ (since $\frac{x^2-4}{x-2}=\frac{(x-2)(x+2)}{x-2}=x+2$ for $x
e2$, then take $x	o2$).

Question 3

State the Intermediate Value Theorem (IVT) and name common types of discontinuity.

Accepted Answer

IVT: If $f$ is continuous on $[a,b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c\in[a,b]$ such that $f(c)=N$. Discontinuities: removable (a “hole”), jump (finite but unequal one-sided limits), infinite (vertical asymptote/unbounded), oscillatory (no limit due to oscillation).

Limits and Continuity flashcards

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