1
Q
When does a limit exist? Prove with K, 𝜖, etc
And at infinity
A
lim f(x) = k
for all e > 0, there exists d > 0 st |f(x)-K| < e for all 0 < |x - x0| < d
at infinity lim f(x) = K
for all e > 0, there exists d > 0 st |f(x) - K | < e for all x > d
2
Q
Definition of big O
A
for f(x), g(x) in R, f(x) = O(g(x)) as x -> a iff there exists constants e, K st |f(x)| ≤ K|g(x)| for all |x-a| < e
3
Q
lim 3x² - 5x / 5x² + 2x - 6
A
3/5
4
Q
lim xsqrt((x+1)) - x
A
lim[x(1+1/2x - 1/8x2 +…) - x] = 1/2
5
Q
lim f(x)^g(x)
A
limf ^ lim g
6
Q
lim (x+a / x-a) ^ x
A
(1+a/x / 1- a/x)^x
lim y->0 (1+ay / 1-ay)^1/y
lim e^(ln 1+ay - ln 1- ay) / y
lim e^(2ay+..)/y
e^2a
7
Q
lim e^f(x)
A
e^lim f
8
Q
L’hopital and when allowed?
A
if both are 0 then f’(x) / g’(x)
9
Q
Definition of continuous
A
f(a) exists and lim x->a f(x) = f(a)
