Chapter 2- Limits Flashcards
Instantaneous rate of change
Limit of average rates of change
Average rate of change
y = f(x) over an interval [x0,x1]
Average rate of change = delta f/delta x =
f(x1) - f(x0)
————–
x1 - x0
Linear function
f(x) = mx + b
Lim f(x) = L x-> c
The limit of f(x) as x approaches c is L
Either limit exists or limit does not exist
Lim = L
x-> c-
If f(x) converges to L as x approaches c through values less than c.
Lim = L
x-> c+
If f(x) converges to L as x approaches c through values greater than c.
One sided limits
Limit only exists if both one sided limits exist and are equal
Infinite limits
Limits can approach infinity or negative infinity
Lim (f(x) + g(x)) =
x-> c
Lim f(x) + lim g(x) x-> c x-> c
Lim k f(x) =
x-> c
k lim f(x)
x-> c
Lim f(x)*g(x) x-> c
Lim f(x) * lim g(x) x-> c x-> c
f(x)
Lim ——- =
x-> c g(x)
lim f(x) x-> c --------- lim g(x) x-> c
Lim [f(x)]^(p/q) =
x-> c
(lim f(x))^(p/q)
x-> c
Assume that f(c) is defined on an open interval containing x = c.
Then f is continuous at x = c if
lim f(x) = f(c) x-> c
If limit does not exist or if it exists but is not equal to f(c)
We say that f has a discontinuity
For a function to be continuous
- f(c) is defined.
- lim f(x) exists.
x-> c - They are equal.
Removable discontinuity
If lim f(x) exists but is not equal to f(c)
x-> c
Jump discontinuity
Occurs if the one-sided limits lim x-> c- f(x) and lim x-> c+ f(x) exist but aren’t equal
Infinite discontinuity
If one or both of the one sided limits is infinite
Substitution method
If f(x) is known to be continuous at x = c, then the value of the lim f(x) is f(c). x-> c
Indeterminate forms
0/0
Infinity/infinity
Infinity*0
Infinity - infinity
If f(x) is indeterminate at x = c
Transform f(x) algebraic ally into a new expression that is defined and continuous at x = c. Then evaluate by substitution.
sin x
Lim ——— =
x-> 0 x
1
1 - cos x
Lim ————- =
x-> 0 x
0
Squeeze Theorem
We say that f(x) is squeezed at x = c if there exists functions l(x) and u(x) such that l(x) < f(x) < u(x) for all x not equal to c in an open interval I containing c, and Lim f(x) = L x-> c
Lim x^n =
x-> infinity
Infinity
Lim x^-n =
x-> -infinity
0
3x^4
Lim ——- =
x-> c 7x^4
3/7
3x^3
Lim ———- =
x-> infinity 7x^4
0
3x^8
Lim ———- =
x-> -infinity 7x^3
-infinity
3x^7
Lim ———- =
x-> -infinity 7x^3
Infinity
Intermediate Value Theorem
Continuous function cannot skip values
If f(x) is continuous on [a,b] with f(a) not equal to f(b), and if M is a number between f(a) and f(b), then f(c) = M for some c E (a,b).
Formal definition of a limit
Learn it with problems in notebook
