Download MTH 125 - Shelton State Community College

SECTION 2.4
Continuity & One-Sided Limits
Discontinuous v. Continuous
Formal Definition of Continuity
Definition of Continuity (p. 90)
Continuity at a Point: A function 𝑓 is continuous at 𝒄 if the following
three conditions are met.
1. 𝑓(𝑐) is defined.
2. lim 𝑓(𝑥) exisits.
𝑥→𝑐
3. lim 𝑓(𝑥) = 𝑓(𝑐).
𝑥→𝑐
Continuity on an Open Interval: A function is continuous
on an open interval (a,b) if it is continuous at each point in
the interval. A function that is continuous on the entire real
line (-∞,∞) is everywhere continuous.
Two Types of Discontinuity
• Removable & Non-Removable
• A discontinuity at 𝑐 is called removable if 𝑓 can be made
continuous by appropriately defining (or redefining) 𝑓 𝑐 .
One-Sided Limits
1.
2.
lim+ 𝑓(𝑥)
𝑥→𝑐
lim− 𝑓(𝑥)
𝑥→𝑐
“the limit of 𝑓 as 𝑥
approaches 𝑐 from the right”
“the limit of 𝑓 as 𝑥
approaches 𝑐 from the left”
When Does a Limit Exist?
Theorem 2.10 The Existence of a Limit (p. 93)
Let 𝑓 be a function and let 𝑐 and 𝐿 be real numbers. The
limit of 𝑓(𝑥) as 𝑥 approaches 𝑐 is 𝐿 if and only if
lim+ 𝑓(𝑥) = 𝐿
𝑥→𝑐
and
lim 𝑓(𝑥) = 𝐿.
𝑥→𝑐 −
Example 1 (#6)
a.
b.
lim+ 𝑓(𝑥) =
𝑥→𝑐
lim− 𝑓(𝑥) =
𝑥→𝑐
c. lim 𝑓(𝑥) =
𝑥→𝑐
Calculating One-Sided Limits
1. Plug in the x-value that you are approaching.
2. If you get a real number, then that’s your limit.
3. If not, try some algebra to see if things can
cancel.
4. If that doesn’t work, then plug in x-values
extremely close to the number you are
approaching or graph it.
Example 2
Find the limit if it exists. If not, explain.
a.
2−𝑥
lim+ 2
𝑥→2 𝑥 −4
b.
𝑥−10
lim +
𝑥→10 𝑥−10
c.
lim +
∆𝑥→0
𝑥+∆𝑥 2 +𝑥+∆𝑥− 𝑥 2 +𝑥
∆𝑥
Example 2 (cont.)
Find the limit if it exists. If not, explain.
a.
b.
𝑥,
lim+ 𝑓(𝑥) where 𝑓 𝑥 =
1 − 𝑥,
𝑥→1
𝑥≤1
𝑥>1
𝑥 2 − 4𝑥 + 6,
𝑥<2
lim 𝑓(𝑥) where 𝑓 𝑥 =
𝑥→2
−𝑥 2 + 4𝑥 − 2, 𝑥 ≥ 2
Continuity on a Closed Interval
Definition of Continuity on a Closed Interval (p. 93)
A function 𝑓 is continuous on the closed interval [a,b] if
it is continuous on the open interval (a,b) and
lim+ 𝑓(𝑥) = 𝑓(𝑎)
𝑥→𝑎
and
lim− 𝑓(𝑥) = 𝑓(𝑏).
𝑥→𝑏
The function 𝑓 is continuous from the right at 𝑎 and
continuous from the left at 𝑏.
Pictorial Representation
Example 3
Discuss the continuity of the function on the closed interval.
a.
𝑓 𝑡 = 2 − 9 − 𝑡2
b.
𝑔 𝑥 =
1
𝑥 2 −4
[−2, 2]
[−1, 2]
Example 4
Find the 𝑥-values where 𝑓 is discontinuous. If any, which
are removable?
3
𝑥−2
a.
𝑓 𝑥 =
b.
𝑓 𝑥 = 𝑥 2 − 2𝑥 + 1
c.
d.
𝑓 𝑥 =
𝑥−3
𝑥 2 −9
𝑓 𝑥 =
𝑥−3
𝑥−3