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One-Sided Limits and Continuity By Dr. Julia Arnold
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One-Sided Limits and Continuity By Dr. Julia Arnold.

Jan 14, 2016

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Page 1: One-Sided Limits and Continuity By Dr. Julia Arnold.

One-Sided Limits and Continuity

ByDr. Julia Arnold

Page 2: One-Sided Limits and Continuity By Dr. Julia Arnold.

One-Sided Limits

The function f has the right-hand limit L as x approaches a from the right written

L)x(flimax

If the values f(x) can be made as close to L as we please by taking x sufficiently close to (but not equal to) a and to the right of a.

The function f has the left-hand limit M as x approaches a from the left written

M)x(flimax

If the values f(x) can be made as close to M as we please by taking x sufficiently close to (but not equal to) a and to the left of a.

Page 3: One-Sided Limits and Continuity By Dr. Julia Arnold.

Theorem 3: Le f be a function that is defined for all values of x close to x = a with the possible exception of a itself. Then

L)x(flimax L)x(flim)x(flim

axax If and only if

Thus the two-sided limit exists if and only if the one-sided limits exist and are equal.

Example 1: Let

0ifx,x

0ifx,x)x(f

)x(flim0xFind Since = -0 =0 both the left and

right limits exist and are equal thus the limit is0.

0

Page 4: One-Sided Limits and Continuity By Dr. Julia Arnold.

Example 2: Let

0ifx,1

0ifx,1)x(g

)x(glim0xFind

1)x(glim0x The

1)x(glim0x

Is the right hand limit.

Is the left hand limit

Since they are unequal the limit of g(x) as x approaches 0 does not exist.

Page 5: One-Sided Limits and Continuity By Dr. Julia Arnold.

Continuous Functions

A function f is continuous at the point x = a if the following conditions are satisfied.1. f(a) is defined.2. exists

3.

)x(flimax

)a(f)x(flimax

If a function is not continuous at a point then it is discontinuous.

Page 6: One-Sided Limits and Continuity By Dr. Julia Arnold.

On the graph we can see that there are 2 points of discontinuity.Let’s see which of the three properties are violated.Is f(A) defined? Yes (the solid dot)Does the exist? No the right limit and left limit are not the same.Is f(B) defined? No

A B

)(lim xfAx

Page 7: One-Sided Limits and Continuity By Dr. Julia Arnold.

We will now look at 3 functions all of which are discontinuous at some point. We will also examine which of the 3 properties is violated.

Equation 1:

1x1

1x2xxf

,

,)(

Since f(x)= x+2 is a straight line and straight lines are continuous, we can conclude that the discontinuity must come from the piecewise definition of the new function and the discontinuity must occur at x = 1.By definition f(1) = 1 but on the line f(1) = 3 which implies that 3xf1x )(lim

Hence, property 1 and 2 are okay.Property 3 is violated which requires that

13but

1fxf1x

...

)()(lim

F(x) is discontinuous at x=1because it violates the 3rd property

Page 8: One-Sided Limits and Continuity By Dr. Julia Arnold.

Equation 2: 2x4x

xf2

)( Since 2x2x

2x2x2x4x2

))((

We can conclude that this is a straight line x+2 but we know that x because of division by 0 thus it is a straight line with a hole in it at x =2. The discontinuity occurs at the domain problem x = 2 and is discontinuous because f(2) is not defined. Violates property 1.

2

Equation 3:

0ifx1

0ifxx1

xg,

,)(The function 1/x is undefined at 0 but this function gives a value for g(0) namely -1 thus property 1 is not violated. The problem point is again the domain problem x = 0, so the question iswhat is the

)(lim xg0x

Since the limit is infinity, the limit doesn’t exist which violates property 2.

Page 9: One-Sided Limits and Continuity By Dr. Julia Arnold.

What type of functions are continuous at every point?A. Polynomial functionsB. Rational functions are continuous everywhere except where the denominator is 0.

Theorem 4: The Intermediate Value TheoremIf f is a continuous function on a closed interval [a,b] and M is any number between f(a) and f(b) then there is at least one number c in [a,b] such that f(c ) = M

Theorem 5: Existence of Zeros of a Continuous FunctionIf f is a continuous function on a closed interval [a,b] and if f(a) and f(b) have opposite signs then there is at least one solution of the equation f(x)=0 in the interval (a,b).

Page 10: One-Sided Limits and Continuity By Dr. Julia Arnold.

x

yFor Th. 4Pick a closed interval on the x axis say [1,2.5]f(1)=3 and f(2.5)=5So if I pick a y value between 3 and 5 (say 4) then there must be a value x between 1 and 2.5 such that f( c) = 4 c = 2.25

Page 11: One-Sided Limits and Continuity By Dr. Julia Arnold.

x

yFor Th. 5Pick a closed interval on the x axis say [-2,0]f(-2)=-8 and f(0)=5Since -8 and 5 are opposite in sign the continuous graph must have crossed the x axis somewhere between -2 and 0. It looks like it could be at-1.5

Page 12: One-Sided Limits and Continuity By Dr. Julia Arnold.

Now go to the homework.