The Formal Definition of the Limit
Department of Mathematics and Statistics
September 13, 2013
Calculus I (James Madison University)
Math 235
September 13, 2013
1/5
The Definition of the Limit
Definition
The limit lim f (x) = L means that for all > 0, there exists δ > 0 such
x→c
that:
if x ∈ (c − δ, c) ∪ (c, c + δ), then f (x) ∈ (L − , L + ).
This is equivalent to saying:
if 0 < |x − c| < δ, then |f (x) − L| < .
Calculus I (James Madison University)
Math 235
September 13, 2013
2/5
The Definition of the Limit
Definition
The limit lim f (x) = L means that for all > 0, there exists δ > 0 such
x→c
that:
if x ∈ (c − δ, c) ∪ (c, c + δ), then f (x) ∈ (L − , L + ).
This is equivalent to saying:
if 0 < |x − c| < δ, then |f (x) − L| < .
Calculus I (James Madison University)
Math 235
September 13, 2013
2/5
The Definition of the Limit
Definition
The limit lim f (x) = L means that for all > 0, there exists δ > 0 such
x→c
that:
if x ∈ (c − δ, c) ∪ (c, c + δ), then f (x) ∈ (L − , L + ).
This is equivalent to saying:
if 0 < |x − c| < δ, then |f (x) − L| < .
Calculus I (James Madison University)
Math 235
September 13, 2013
2/5
Limits are Unique
Theorem (Uniqueness of Limits)
If lim f (x) = L and lim f (x) = M, then L = M.
x→c
x→c
Calculus I (James Madison University)
Math 235
September 13, 2013
3/5
Limits are Unique
Theorem (Uniqueness of Limits)
If lim f (x) = L and lim f (x) = M, then L = M.
x→c
x→c
Calculus I (James Madison University)
Math 235
September 13, 2013
3/5
One-Sided Limits
Calculus I (James Madison University)
Math 235
September 13, 2013
4/5
One-Sided Limits
The left limit lim f (x) = L means that for all > 0, there exists δ > 0
x→c −
such that:
if x ∈ (c − δ, c), then f (x) ∈ (L − , L + ).
Calculus I (James Madison University)
Math 235
September 13, 2013
4/5
One-Sided Limits
The left limit lim f (x) = L means that for all > 0, there exists δ > 0
x→c −
such that:
if x ∈ (c − δ, c), then f (x) ∈ (L − , L + ).
The right limit lim+ f (x) = L means that for all > 0, there exists δ > 0
x→c
such that:
if x ∈ (c, c + δ), then f (x) ∈ (L − , L + ).
Calculus I (James Madison University)
Math 235
September 13, 2013
4/5
One-Sided Limits
The left limit lim f (x) = L means that for all > 0, there exists δ > 0
x→c −
such that:
if x ∈ (c − δ, c), then f (x) ∈ (L − , L + ).
The right limit lim+ f (x) = L means that for all > 0, there exists δ > 0
x→c
such that:
if x ∈ (c, c + δ), then f (x) ∈ (L − , L + ).
For a Limit to Exist, the Left and Right Limits Must Exist and be
Equal
The limit lim f (x) = L if and only if lim f (x) = L = lim+ f (x).
x→c
Calculus I (James Madison University)
x→c −
Math 235
x→c
September 13, 2013
4/5
Limits involving Infinity
Calculus I (James Madison University)
Math 235
September 13, 2013
5/5
Limits involving Infinity
The infinite limit lim f (x) = ∞ means that for all M > 0, there exists
x→c
δ > 0 such that:
if x ∈ (c − δ, c) ∪ (c, c + δ), then f (x) ∈ (M, ∞).
Calculus I (James Madison University)
Math 235
September 13, 2013
5/5
Limits involving Infinity
The infinite limit lim f (x) = ∞ means that for all M > 0, there exists
x→c
δ > 0 such that:
if x ∈ (c − δ, c) ∪ (c, c + δ), then f (x) ∈ (M, ∞).
The limit at infinity lim f (x) = L means that for all > 0, there exists
x→∞
N > 0 such that:
if x ∈ (N, ∞), then f (x) ∈ (L − , L + ).
Calculus I (James Madison University)
Math 235
September 13, 2013
5/5
Limits involving Infinity
The infinite limit lim f (x) = ∞ means that for all M > 0, there exists
x→c
δ > 0 such that:
if x ∈ (c − δ, c) ∪ (c, c + δ), then f (x) ∈ (M, ∞).
The limit at infinity lim f (x) = L means that for all > 0, there exists
x→∞
N > 0 such that:
if x ∈ (N, ∞), then f (x) ∈ (L − , L + ).
The infinite limit at infinity lim f (x) = ∞ means that for all M > 0,
x→∞
there exists N > 0 such that:
if x ∈ (N, ∞), then f (x) ∈ (M, ∞).
Calculus I (James Madison University)
Math 235
September 13, 2013
5/5