what is a limit ? when does a limit exist? continuity discontinuity types of discontinuity
Post on 19-Dec-2015
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What is a limit ?
When does a limit exist?
Continuity Discontinuity
Types of discontinuity
What is a limit?A limit of a function is an intended
“height” of a function at a point
Lim f (x) = 4
X 2
When does a limit exist?
When does a limit exist?f (x) = x²
When does a limit exist?
When does a limit exist?
With a plus sign
When does a limit exist?
With a minus sign
In order to have a limit at a point
At that point:
Left limit MUST EQUAL Right limit
Lim f (x)= 1 Lim f (x)= 1
X 4ˉ X 4
Lim f (x)= 1
X 4With no sign
+
Limit exist
Lim f (x)= 1 X 4
Infinite Limits( are not limits)
How to evaluate a limit, in case of algebraic functions?
1. Finding the value of the function at the point (Substitution in the formula of the function) If the function is continuous at the point
2. Factoring (The case 0/0 )
3. The Conjugate Method (The case 0/0 )
Substitution
Factoring
Factoring
The Conjugate MethodWhat is a conjugate?
X-16 = (√x - 4) (√x + 4)
AND(√x + 4) is the conjugate of (√x - 4)respect to X-16
(√x - 4) is the conjugate of (√x + 4)respect to X-16
The Conjugate Method
Continuity
f is continuous at a if:
)()(lim afxfax
Example of a function f which is discontinuous, but continuous from the right.
lim f (x) = 4 = f (2)X2+
Example of a function f which is discontinuous, but continuous from the left.
lim f (x) = 4 = f (2)X2ˉ
Discontinuity
A function is discontinues at a if the limit at a is not equal to the value f (a)
A continuous function on R should have:1.no breaks in the graph2.no holes 3.no jumps
Everywhere Continuous Function
24)(lim
xxf
24)(lim
xxf
2
4
* Since f(2) is also equal to 4; then = f(2)
• The function is continuous at 2.
• And since it’s also continuous at all other point’s in R; then it’s everywhere continuous.
Type of discontinuity
1. Removable discontinuity
2. infinite discontinuity
3. jump discontinuity
Jump discontinuity
7)(lim1
x
xf
4)(lim1
x
xf
existnotdoesxfx 1
)(lim
1
7
4
f is continuous on the intervals (-∞ ,1 ) and [1, ∞ )
Infinite discontinuity
4)(lim3
xfx
)(3
xfLimx
3
4
The function is continuous on the interval (-∞ , 3] and (3, ∞ )
Notice that the function is continuous from the left at 3
Removable discontinuity: the limit of f exists at the point but f is not equal to the value of f at
that point.
4)(lim3
x
xf
4)(lim3
xfx
4)(lim3
xfx
3
Notice that the function is neither continues from the left nor continuous from the right at 3
The function is continuous on the interval (-∞,3) and (3,∞)
4
Let
1
2
3
-2
(1,3)
f is not defined at 1 but exists and equal to 3)(lim1
xfx
}1{;2)( Rxxxf
Removable discontinuity: the limit of f exists at the point but f is not defined at that point.