Fichas
C2: Definitions
Fichas del conjunto (26)
- 1Limit Assumption (Intuitive)Suppose f : A ⊂ R → R is a function sufficiently well defined near
x = c, but not necessarily defined at x = c) and L± ∈ R - 2Limit (intuitive part 1)If the values of f(x) approach L+ (L−) as x approaches c from the right (or left), denoted by x → c+ (x → c−), we say that L+ (L−) is the right- (or left-) hand limit of f at x = c and we write
f(c±) = lim
x→c± f(x) = L− or f(x) → L± as x → c± - 3Limit (intuitive part 2)If both left and right limits exist at x = c and they are equal f(c+) = f(c−) = L. Then we say that L is the limit of f at x = c and write
lim
x→c
f(x) = L or f(x) → L as x → c - 4Limit (Intuitive part 3)otherwise, if f(c+) = f(c−), or f(x) → ±∞ as x → c, we say the limit does
not exist. - 5Addtional intuition limit noteThat f(c+) = f(c−) = L does NOT mean that f(c) = L, since f need not be defined at x = c.
- 6Limit (Formal idea)If f is a function defined in some deleted neighbourhood of c, then limit limx→c f(x) = L means for all (∀) ε > 0 there exists (∃) δ(ε) > 0 such that if
0 < |x − c| < δ then |f(x) − L| < ε - 7Limit (Formal notation)Given ε > 0 (no matter how small) we can always find a δ(ε) > 0 such that when x ∈ Nδ(c)\{c} = (c − δ,c) ∪ (c,c + δ) then f(x) ∈ Nε(L) = (L − ε,L + ε) In general, as ε is made smaller, we have to go ever nearer the point x = c in order
to satisfy the condition L − ε < f(x) < L + ε, which is why δ = δ(ε). - 8Standard results for delta (check notes of proof)For a = m = 0 we see that for all choices of δ |f(x) − L| = 0 < ε
For a = 0 we choose δ = ε/|m| then as required
0 < |x − c| < δ = ε/|m|
If m = −2ac then choose δ =
√ε/|a| then as required
0 < |x − c| < δ =
√
ε/|a| - 9One sided limits (left)If f is a function, sufficiently well defined near x = c, then the
(i) left limit limx→c− f(x) = L means ∀ε > 0, ∃ δ(ε) > 0 such that if
x ∈ (c − δ,c) then |f(x) − L| < ε or f(x) ∈ Nε(L) = (L − ε,L + ε) - 10One sided limits (right)If f is a function, sufficiently well defined near x = c, then the
(ii) right limit limx→c+ f(x) = L means ∀ε > 0, ∃ δ(ε) > 0 such that if
x ∈ (c,c + δ) then |f(x) − L| < ε or f(x) ∈ Nε(L) = (L − ε,L + ε) - 11A Vertical AsymptoteSuppose that f is a function sufficiently well defined near x = c. If the value of f(x) increases/decreases without bound as x approaches c from the left or right, then we can write
lim
x→c+
f(x) = ±∞ or lim
x→c−
f(x) = ±∞ or lim
x→c
f(x) = ±∞ - 12Horizontal/slant AsymptoteIf the function f is defined for all |x| > a we say that f has a slant asymptote at the line y = mx + L if one of both of the following hold
lim
x→−∞
f(x)/mx + L= 1 or lim
x→∞
f(x)/mx + L= 1
For the case m = 0 we say f has a horizontal asymptote at y = L. - 13Limit at infinity (Formal definition)If f is a function defined for all x ∈ [a,∞) for some a > 0, then the limit at infinity limx→∞ f(x) = L means that for all (∀) ε > 0 (no matter how small) there exists (∃) an N(ε) > a such that if
x > N then |f(x) − L| < ε - 14Infinite Limit (Formal Definition)If f is function defined in a deleted neighbourhood of c, then the infinite limit limx→c f(x) = ±∞ means that for all (∀) M > 0 (no matter how large) there exists (∃) δ(M) > 0 such that if
0 < |x − c| < δ then |f(x)| > M - 15Quotient limit form 1If limx→c f(x) = L and g(x) → M as x → c then we say that
limx→c f(x)/g(x) is of the form L
M - 16Quotient Limit form 2If limx→∞ f(x) = L and g(x) → M as x → ∞, then we say that
limx→∞ f(x)/g(x) is of the form L
M - 17eThe exponential number e is defined by the indeterminate limit form 0∞ as
e = lim
h→0
(1 + h)^1/h ≈ 2.71828 - 18ContinuousLet f be a function defined on Nδ (c) and f(c) defined, then
(i) f is said to be continuous at x = c if f (c±) = limx→c± f(x) = f(c). - 19Continuous from the LeftLet f be a function defined on Nδ (c) and f(c) defined then
(ii) f is continuous from the left at x = c if f(c−) = limx→c− f(x) = f(c). - 20Continuous from the rightLet f be a function defined on Nδ (c) and f(c) defined, then
(iii) f is continuous from the right at x = c if f(c+) = limx→c+ f(x) = f(c). - 21Important 3 elements of continuityThus continuity requires
(i) f(c) to be defined,
(ii) f(c±) = limx→c± f(x) to exist, and
(iii) f(c±) = limx→c± f(x) = f(c), - 22DiscontinuousIf a function f is not continuous x = c, we say f is discontinuous at x = c, or f
has a discontinuity at x = c. - 23Removable DiscontinuitySuppose f is discontinuous at x = c, then we say that x = c is a
(a) removable discontinuity f(c±) both exist, are equal, but not equal to f(c). - 24Jump DiscontinuitySuppose f is discontinuous at x = c, then we say that x = c is a
(b) jump discontinuity if f(c±) both exist but are not equal. - 25infinite DiscontinuitySuppose f is discontinuous at x = c, then we say that x = c is a
(c) infinite discontinuity if either limx→c− f(x) or limx→c+ f(x) are infinite. - 26Interval ContinuityA function f is continuous on an interval I ⊂ R, denoted by f ∈ C [I], if f is continuous at all points in the interior, right continuous at any closed left end point, and left continuous at any closed right endpoint.
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- (Chapter 2: Limits & Continuity) Calculus Fichas sobre C2: Definitions, creado por Luke Danaher el 26/12/2016.
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