Limits and continuity

Question 1

limit of a function general idea

Answer 1

a function f(x) has a limit L at x = a if f(x) is close to L whenever x is close to a
if we have an acceptable error (ε >0) between f(x) + L, need |f(x) -L| < ε for all x in some interval around a
if for any ε >0 i can find a distance d >0 such that
|f(x) - L| < ε for all 0<|x-a|<d
then f(x) has a limit L as x tends to a

Question 2

limit definition

Answer 2

f(x) has a limit L as x tends to a if for all ε >0 there exists d >0, such that |f(x) -L| < ε for all x such that 0 < |x-a| < d
we write lim f(x) as x -> a = L or f(x) -> L as x-> a
if there is no L the limit does not exist

Question 3

continuity

Answer 3

a function f(x) is continuous at x = a if:
- f(a) exists
- lim f(x) as x->a exists
- lim f(x) as x->a = f(a)

Question 4

function continuous on a subset S of its domain

Answer 4

if its continuous at every point in S
continuous if its continuous at every point in its domain

Question 5

right sided limit

Answer 5

L + = lim f(x) as x tends to a+ (tends to a from above) if for all ε >0 there exists d >0, such that |f(x) -L+| < ε for all x such that 0 < x-a< d
x> a

Question 6

left sided limit

Answer 6

L- = lim f(x) as x tends to a- (tends to a from below) if for all ε >0 there exists d >0, such that |f(x) -L-| < ε for all x such that 0 < a-x < d
x<a

Question 7

limit exists if

Answer 7

if and only if L+ and L- both exist and are =
L = L+ = L-

Question 8

3 types of discontinuity

Answer 8

• removable discontinuity
• jump discontinuity
• infinite discontinuity

Question 9

removable discontinuity

Answer 9

L exists but f(a) =/= L
we can always remove this to make a continuous function
g(x) is
f(x) if x =/=a or L if x = a

Question 10

jump discontinuity

Answer 10

L + and L- exists but L+ =/= L-
(so L does not exist)

Question 11

infinite (or essential) discontinuity

Answer 11

at least one of L+ or L- does not exist

Question 12

facts about limits

Answer 12

• the limit is unique (at each point)
• if f(x) = g(x) (except possibly at x = a) in some open interval containing a then lim f(x) as x tends to a = lim g(x) as x tends to a
• if f(x) >= k on either some interval (a,b) or (c,a) and if lim f(x) as x -> a = L L>= k (opposite if f(x) <= k)
• calculus of limits theorem (COLT) if lim f(x) = L and lim g(x) = M then
  1) lim(f(x)+ g(x)) = L + M
  2) lim( f(x) . g(x) ) = L.M
  3) if M=/=0 lim(f(x)/g(x)) = L/M

Question 13

facts about continuity

Answer 13

• if f and g are continuous then so are f+g, fg, f/g, and |f|
• all polynomial, rational, trig and hyperbolic functions are continuous
• if lim g(x) = L and f(x) is continuous at x = L then lim fºg(x) = f(L)

Question 14

pinching / squeezing theorem

Answer 14

If g(x) <= f(x) <= h(x) for x =/= a in some open interval containing a and if lim g(x) = lim h(x) = L
then lim f(x) = L

Question 15

2 trig limits

Answer 15

lim sinx/x = 1 as x -> 0
lim (1-cos x)/x = 0 as x -> 0

Question 16

limits to infinity rough idea

Answer 16

a function as a limit to infinity if f(x) can be kept arbitrarily close to L by making x sufficiently large

Question 17

limit to infinity

Answer 17

f(x) has a limit L as x -> ♾️ if for all epsilon >0 there exists S >0 such that |f(x) - L < epsilon for all x >S
(same for -♾️ but x<S)

Question 18

method for limit to infinity

Answer 18

make substitution u = 1/x then tend u -> 0+ for f(1/u)

Question 19

asymptote and limits

Answer 19

the graph of f(x) has a horizontal asymptote to the right (or left) at y = L if the lim x->♾️ f(x) = L (or lim x -> -♾️ f(x) = L)

Question 20

intermediate value theorem (IVT)

Answer 20

if f(x) is continuous on [a,b] and u is any value between f(a) and f(b)
then there exists c ∈ (a,b) such that f(c) = u

Question 21

sgn(x)

Answer 21

1 for x>0
0 for x = 0
-1 for x<0

Question 22

application of IVT

Answer 22

is finding roots of a function
if f is continuous on [a,b] and f(a) <0 <f(b) or f(b)<0 <f(a) then by IVT theres at least one root such that f(x) = 0 between a and b