1.4 Continuity & Limits

Ms. Hernandez

Calculus

Calculus

• 1.4 Continuity and 1-sided limits

• Student objectives:– Understand & describe & find continuity at a

point vs continuity on an open interval– Find 1-sided limits – Use properties of continuity– Understand & use the Intermediate Value

Theorem

Continuity

• Continuous line

• Trace ur finger along the line

• If there are no breaks along the line

• No jumping allowed!

• No breaking allowed!

• No gaps allowed!

• Line = function = f(x)

Continuity example

• P68, see fig 1.25 – 3 graphs

• Discuss w/group– Why are the 3 graphs not continuous at x = c?

Graph 1 – gap in the graphfunction is not defined at x = cGraph 2 – break in the graphthe lim of f(x) dne at x = cGraph 3 – jump in the graphthe lim of f(x) exists at x=c but does not equal f(c)

Def of Continuity

Continuous at a point if:1. function is defined at x = c2. the lim of f(x) at x = c3. the lim of f(x) exists at x=c and equals f(c)

p68

Def of Continuity

• Continuity on an open interval

• (a,b) some points on a graph or f(x)

• Interval includes everything in between a & b

• So that’s why we say on an open interval

• But what is that?

Def of Continuity

• If you can trace your finger all along the line on the interval (a,b) then its conts.

• Include a & b.• So no gaps, jumps, or breaks.• If there is a gap/jump/break then f(x) has a

discontinuity– Removable discontinuity (can b fixed easily)

• P69 Fig 1.26(a),(c)

– Non-removable discontinuity (not 2 easy)• P69 Fig 1.26 (b)

Continuity of a function

• Discuss example 1 pg 69– How does the domain clue you into f(x)

continuity?

Not defined then its not conts there!

Vertical asymptotes

where domain is undefined for f(x)

1-sided limits

• Limit from the left (-)

• Limit from the right (+)

• When working with square roots take from the +

• p70

Continuity on a closed interval

• A function is continuous on the closed interval if

• Lim from left exists

• Lim from right exists

• Ex 4 pg 71

Charles’s Law & Absolute Zero

• Read over example 5 on pg 72

• What steps would you take to find absolute zero on the Fahrenheit scale

• What property must the temp scale have, according to Charles’s Law

Thm 1.11 Properties of Continuity

• p73– Scalar– Sum & difference– Product– Quotient– All polynomial (in their domain)– All rational (in their domain)– All radical (in their domain)– All trig (in their domain)– So how many functions are conts at every point in their

domain?

Consequences of Continuity

• Composite functions

• Thm 1.12 if 2 functions f, g are conts then their composites are also conts

• What’s the consequence of the limit of this composite function?

• Is this any different from last lecture?

• If so, how?

Testing for continuity

• Describe intervals that f(x) is continuous on

• Squeeze thm & continuity

IVT: Intermediate Value Theorem

• Tells you why/what but not HOW

• Existence theorem

• If f is conts on the closed interval [a,b] and k is any number between f(a) and f(b) (ON THE GRAPH so y = k somewhere between endpoints a

& b) then there is at least one number c (that

we can input into function) in [a,b] such that f(c) = k.

APPLY IVT

• Use the IVT to show that the polynomial function f(x) = x2 + 2x –1 has a zero in the interval [0,1].– We know f(x) is continuous on [0,1] b/c

• f(0) = 02 + 2(0) –1 = -1 -1 is on the graph– Testing left endpoint

• f(1) = 12 + 2(1) –1 = 2 2 is on the graph– Testing right endpoint

– f(0) < 0 -1 < 0 – f(1) > 0 2 > 0– So zero is on the graph in between the endpoints [a,b]– So then some number c could be plugged into f(x) and get 0

since 0 in between f(a) and f(b) or [a,b].– See Ex 8 on pg 75

Some examples

1.4 #3-6. I can tell from the graph that f(x) is not conts at x = c. If I ask you on a non-calculator quiz/test to verify continuity of f(x) how would you do that?

We must examine the 3 parts of the definition of continuity. So which part of the continuity theorem is f(x) failing?

#3, #4

• Is f(c) defined? Yes f(c) is defined.

• Does the limit exist at f(c)? Yes, it does! The left and right limits are equal.

• Does the limit value at c = f(c)? Nope!

#5,#6

• Is f(c) defined? Yes f(c) is defined.

• Does the limit exist at f(c)? Nope, left and right limits are not equal.

• STOP.

1.4 #9• lim x -3- = ?• Vertical asymptotes always occur at points where the domain is not defined• http://www.purplemath.com/modules/asymtote.htm

• lim x -3- = DNE b/c grows without bound• Does your graphing calculator help?

1.4 #41

• Domain does not include –2 or 5• Lim x 5 DNE

– Non-removable discontinuity– Indeterminate form

• Lim x -2 = -1/7– Removable discontinuity– Even though –2 is not in the domain, (x+2) can be

factored out and you get an answer of –1/7 for the limit

– Does your calculator help you?

1.4 # 57) Find constants a & b such the the function is conts on the entire real line

3

2

, 2( )

, 2

x xf x

ax x

f(2) = 8

So I need to find a such that

Because then my piecewise function would be continuous.

2

2lim 8x

ax

22 2

8 88 2

2ax a a

x

3

2

, 2( )

2 , 2

x xf x

x x

In solving for a, I set it equal to the limit value I need. When I calculate a limit I would like to direct sub, so I plug in 2. Solving, a must be 2.

1.4 #75 Explain why the function has a zero in the specified interval

• f(x) is continuous on what interval?• Evaluate the endpoints of the interval?• Is zero on the graph in between the endpoints?

4 31( ) 3

16f x x x is continuous on the interval [1,2]

f(1) = 33/16

f(2) = -4

So f(c) = 0 works and zero is in the interval for at least one value of c between 1 and 2.

1.4 #100-103

• 100 – True all conditions are metf(c) = L is defined, lim xc = L exists, and f(c) = lim xc

• 101 - True b/c one of the limits does not equal the function at x = c

• 102 – False b/c a rational function can be written as P(x)/Q(x) where P and Q are polynomials of degree m and n, respectively. So it will at most n discontinuities.

• 103 - False b/c f(1) is not defined and lim x1 DNE