basic definition of limits
the limit of a function is ___ (y-value) as x approaches to __ (x-value) from the left and right side
if lim as x-> a :
- can it be defined at point a?
not necessarily
Thus: limit may exist even if f(a) is undefined
when does a limit NOT EXIST
when the values of a limit from the right side is different from the left side
if lim (fx) exists, then it is ___?
unique
Example of an equation wherein the function value is undefined but the limit exists
f(x) = [(x^2 -9) / (x-3)]
Cite the 8 Rules on Evaluating limits
- Constant
- Number A
- Coefficient of f(x)
- Addition/Subtraction
- Multiplication
- Division
- Power
- Power Root (n = even; f(x) ≠ 0)
For equations like lim as x-> 1 =
_/x - 1
———-
x - 1
how to solve
rationalize
dont simplify muna
=1/2
if a given is
lim f(x) as x -> 1^( - ) = ?
lim f(x) as x -> 1^( + ) = ?
what are the:
1. Possible answers
2. What do you call these functions
3. Exceptions to these
- or - infinity
- One-sided Limits
- Piecewise Functions
4 RULES for limits involving infinities + RECHECKING
- Addition/Subtraction: Highest Degree
- Division: Bottom Heavy = 0
- Division: Top Heavy = Cross out other insignificant value in the numerator then the denominator
- Division: Equal degree in numerator and denominator = divide coefficients
RECHECKING
divide both numerator & denominator by 1/(x^highest degree in denominator)
Find out whats different then solve:
lim as x->3
1 ——— (x - 3)^2
THIS IS NOT A ONE SIDED FUNCTION
since = 1/0
infinity ang sagot
LIMITS
if k is a constant and:
k ^ ∞
what is the answer
∞, if k>1
0, if 0<k<1
3 rules for a continuous function
- f(x) exists
- lim exists
- f(x)=lim
CONTINUOUS FUNCTION
if piecewise eq what to do
if not piecewise
piecewise: for LIM, check right and left side
not piecewise: for LIM, directly substitute na
what to do
- lim ln f(x)
- lim sin f(x)
- lim e^(f(x))
- ln (lim f(x))
- sin (lim f(x))
- e^(lim f(x))
CONTINUITY FUNCTIONS IN
- open intervals
- closed intervals [a, b]
- all numbers in between must be continuous
- a. all numbers in between mus be continuous
b. lim x-> a+ = f(a)
c. lim x-> b- = f(b)
FACTORING RULES
1. a2 - b2
- (a+b)2
- (a-b)2
- a3 + b3
- a3 - b3
- (a+b) (a-b)
- a2 +2ab +b2
- a2 -2ab +b2
- (a + b) (a2 - 2ab + b2)
- (a - b) (a2 + 2ab + b2)
What to remember in one sided functions?
- Factor
- Substitute x
- Analyze
SUBSTITUTE X KASI approach x na sya,…
and look at if from the right/left
RELATIONSHIP STATEMENT between function continuity and differentiation
If the function f(x) is differentiable at x = c, then it is also continuous at x = c.
If possible, give an example of the following:
a. A function which is differentiable and continuous.
b. A function which is not continuous and not differentiable.
c. A function which is continuous but not differentiable.
a. x^2
b. g(x) =
1 if | x < 0
0 if | x ≥ 0
c. |x|
General equation for derivatives
f’(x) = lim f(x+h) - f(x)
x->h —————–
h
11 Rules of Differentiation
- Constant
- Power
- Coefficient
- Sum & Difference
- Product [ g(x) h’(x) + g’(x) h(x) ]
- Quotient [ wherein g/h: {h(x) g’(x) - g(x) h’(x)} / h(x)^2}
- e^x
- ln (1/x)
- log (1/xln10)
- sin(x) (cos(x))
- cos(x) (-sin(x))
What is the geometric and physical interpretation of f’(x) = a
geometric: slope of the tangent line to the graph of f(x) at a
physical: - instantaneous rate of change, velocity
What are the derivatives of the ff:
1. f(x) = e^π
2. f(π) = e^x
3. f(x) = 4x^2e^x
- 0 kasi constant, π is not x
- e^π
- Product Rule: 4e^x (2x+x^2)
MEANING OF:
dy/dx
y = f( )
x = x
