Post on 23-Feb-2016
description
• Advise from last year’s crew:– Do the homework (even if it’s not graded!)– Check the blog– Buy an extra prep book– Keep a 3 ring binder to stay organized– Stick with it– Stay for extra help– Bring a calculator every day– Calculator sections are not a joke – prepare for
them!• Let’s get started!
Limits (1.2 and 1.3)
What happens to as x approaches 1?1,11)(
3
xxxxf
Limit• A limit is a y-value that the graph of a function
approaches as x gets closer and closer to a particular value from either direction
• Limits can be used more generally to describe the behavior of a function near a particular x-value
• Notation: Lxfcx
)(lim
Does a limit necessarily exist?
No.- The function might approach a
different y-value from one side verses the other.
- If there is a vertical asymptote at x = c, then the function is going to ±∞
Assuming the limit does exist, how can you find it?
1. Try direct substitution (review your unit circle!)2. If that results in the indeterminate form , then
the limit still exists and you can find it by…* Using the table from your calculator* Factoring out a common term.
Example:
00
11)( 2
xxxf
)(lim2
xfx
)(lim1
xfx
)(lim1
xfx
The Big Ideas
• When we talk about limits, we don’t necessarily care what’s happened to f and x = c, but right around x = c.
• Limits are the foundation for everything we talk about in calculus.
One-Sided Limitsrefers to the limit from both sides
of the function around c
refers to the limit as x approaches c from the left.
refers to the limit as x approaches c from the right.
)(lim xfcx
)(lim xfcx
)(lim xfcx
How can we use one-sided limits to determine whether the limit of a piecewise function as x approaches a “change –over” value occurs?
),2cos(
,12,3
)(
2
xxx
xf222
xxx
Special Limits
Just know them….
1sinlim0
x
xx
axax
x
)sin(lim0
0cos1lim0
xx
x
Continuity (1.4)There is a strong connection between limits and continuityA function f(x) is continuous at x = c if…
1. must exist
2. must exist
3.
A function is continuous if you can draw its graph without picking up your pencil!
)(cf
)(lim xfcx
)()(lim cfxfcx
Which functions are always continuous?
- Polynomial (constant, linear, quadratic, cubic..)
- Sine, cosine- Absolute value- Radical functions of odd degrees- Exponential
Which functions are continuous, but have a restricted domain?
- Radical functions of even degrees- Logarithmic functions- Rational functions
Which functions are not always continuous?
- Tangent, secant, cosecant, cotangent- Piecewise *
Investigating Continuity
166)(.1 4
2
x
xxxf
,4
,1)(.2
2
2
x
xxf
11
xx
,4
,0
,2
)(.33 x
x
xf111
xxx
Example
Find the value of a and b so that the following function is continuous at x = 1
Vertical Asymptotes (1.5)
If you plug c into f(x) and get… f has a removable discontinuity at x = c (hole)
and exists.
f has vertical asymptote at x = c and DNE. (nonremovable discontinuity)
00
)(lim xfcx
0a
)(lim xfcx
Big IdeaEven though limits around vertical asymptotes don’t
exist, we can use them to describe the behavior of the graph on either side.
Examples on notes
Special Function .
• This function has a jump discontinuity at x = 0. • What are the various limits?• Look at another example:
xx
xf )(
5102
)(
xx
xf
Horizontal Asymptotes (3.5)
13)( 2
2
xxxf
Definition of H.A.
The graph of a function f(x) has a horizontal asymptote at y = a if, as x gets infinitely large in either or both directions, the graph gets closer and closer to the line y = a
Big Idea
• We can use limits to describe end behavior• Not all functions have a finite end behavior
What would be an example of a function whose graph had no horizontal asymptote?
2)( xxf
3)( xxf
What would be an example of a function whose graph has one horizontal asymptote?
xexf )(
5)(
xxxf
What would be an example of a function whose graph has two horizontal asymptotes?
xxf 1tan)(
Note
The graph of a function can still cross over the line y = k and still
Example:
kxfx
)(lim
xxxf sin)(
SummaryIf the higher degree polynomial is….1. In the denominator, then and the H.A.
is along the x – axis2. In the numerator, then D.N.E. (no H.A.)3. A tie between the numerator and denominator, then
the H.A. can be found in the leading coefficients.
0)(lim
xfx
)(lim xfx
baxf
x
)(lim
Examples 1
Examples 2
Slowest to fastest:• Logarithmic• Polynomials
(grow fastest by degree)• Exponentials• Factorials• xxxf )(