Limits and Continuity
Pretty cheap
training
Limits
“Intuition”
Either
Exist
or not
► If they exist,
… you may know what they are
… you may not
► If they do NOT exist,
… you may know that
… you may not!
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
Limits don’t exist
No Limit 
► If there is a ‘jump discontinuity’ (pit?)
… left sided limit and right sided limit unequal
► If there is a vertical asymptote (wall)
… unbounded behavior
► If there is an oscillation (boing boing)
... doesn’t get close to anything (and stay there)
► If end behavior tends to
(zoom!)
… unbounded behavior
► Note that removable discontinuities have limits
… everywhere!
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
Limits don’t exist
No Limit 
► If there is a ‘jump discontinuity’
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
Limits don’t exist
No Limit 
► If there is a vertical asymptote (wall)
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
Limits don’t exist
No Limit 
► If there is an oscillation (boing boing)
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
Limits don’t exist
No Limit 
► If end behavior tends to
(zoom!)
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
Limits don’t exist
No Limit 
► Note that removable discontinuities have limits
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
Limits that exist
Limits !
► Discovery phase
… graphically
… from a table
… any others?
► Verification phase
… calculate
… sometimes it’s just difficult
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
Calculating limits
Limits !
► Basic rules
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
What you need to know – AP AB
Limits !
► Substitute!
► Rational Functions
- Factor! (eliminate 0/0)
- Divide by highest degree x term
► Piecewise Functions (left side vs. right side)
► Radicals
- rationalize
- multiply by ‘conjugate’
► “Rational Radicals to infinity”
watch |x| and signs
► Trig identities
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
Limits – things to keep in mind
Limits !
►
… polynomials taught us everything we feel is “right”
► except for the (*&%$& denominator, you could say the
same about rational functions
► One-sided vs. two-sided limits
► Limits at infinity (end behavior)
► Remember
so don’t be afraid
► Do not manipulate infinity like it’s a number
Be respectful of the subtleties
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
Limits – how to calculate them …
Limits !
► Substitute!
lim f ( x)  f (a)
xa
for every continuous function f (whatever that means)
► Rational Functions
* numerator, denominator not 0 => see substitute!
* numerator not 0; denominator 0 => unbounded
* numerator AND denominator 0 => try to factor
and cancel
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
Limits – how to calculate them …
Limits !
► Piecewise Functions
does limit from right = limit from left?
► Radicals
The most common problem is of the form
infinity – infinity
Try – Multiply top and bottom by conjugate
Create a fraction first if necessary
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
Limits – how to calculate them …
Limits !
► Rational Functions to Infinity
another technique is to divide the top and bottom
by ‘variable to the highest degree’
idea: all but 1 or 2 pieces go to zero!
► Rational Radicals to Infinity (bob-ism)
Sometimes conjugates help
Usually you incorporate dividing by same thing
BUT!
Remember that you leave the sign outside when
you bring things under the radical sign
© 2008, Bob Wilder
Lim to
negative
infinity
Limits and Continuity
Pretty cheap
training
Continuity!
The nice
property
► Definition is critically important (5% of AP test)
A function f is continuous at point x=a
=>
lim f ( x )
iff
exists
x a

f is defined at x=a

lim f ( x) = f(a)
Need all 3
conditions
(why?)
x a
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
Continuity!
► Rigorous definition of limit (and continuity) was an
extremely important development
► From what I remember …
The difficult problems are functions defined
piecewise
AND
you need to find the value of k that makes function
continuous
© 2008, Bob Wilder
Limits and Continuity
Pretty cheap
training
Trigonometric stuff
► You gotta know
sin( x)
lim
1
x 0
x
► x is in radians
► it follows that
1  cos( x)
lim
0
x 0
x
► You gotta know sin2
+ cos2 = 1
© 2008, Bob Wilder