Definition of Limit:
If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f(x) as x approaches c is L. This is written as
Lim f(x) = L
x ->c
(x approaches c, as y approaches L)
Eample:
f(x) = 2x-3
Lim f(x)
x->4
2(4) - 3
8 - 3 = 5
Lim f(x) = 5
X->4
Conditions Under Which Limits Do Not Exist:
(The limit of f(x) as x-> c does not exist if any of the following conditions is true.
1. f(x) approaches a different number from the right side of c than from the left side of c
2. f(x) increases or decreases without bound as x approaches c
3. f(x) oscillates between two fixed values as x approaches c
Properties of Limits:
(Let b and c be real numbers and let n be a positive integer)
lim b = b
x-> c
lim x = c
x->c
lim x^n = c^n
x->c
Techniques for Evaluating Limits
Evaluating Limits by Direct Substitution:
Lim (x^2 + x – 6)
x-> -1
substitute -1 in for x
(-1)^2 + (-1) – 6 = -6
Dividing out Technique:
Lim (x^2 + x – 6/x +3)
Begin by factoring the numerator and dividing out any common factors
Lim (x^2 + x – 6/x +3) = lim (x -2)(x +3)/(x + 3)
= lim (x -2)
= ((-3) – 2)
= -5
This procedure for evaluating a limit is called the dividing out technique. The validity of the procedure stems from the fact that if two functions agree at all but a single number c, they must have identical limit behavior at x = c.
Rationalizing Technique:
If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f(x) as x approaches c is L. This is written as
Lim f(x) = L
x ->c
(x approaches c, as y approaches L)
Eample:
f(x) = 2x-3
Lim f(x)
x->4
2(4) - 3
8 - 3 = 5
Lim f(x) = 5
X->4
Conditions Under Which Limits Do Not Exist:
(The limit of f(x) as x-> c does not exist if any of the following conditions is true.
1. f(x) approaches a different number from the right side of c than from the left side of c
2. f(x) increases or decreases without bound as x approaches c
3. f(x) oscillates between two fixed values as x approaches c
Properties of Limits:
(Let b and c be real numbers and let n be a positive integer)
lim b = b
x-> c
lim x = c
x->c
lim x^n = c^n
x->c
Techniques for Evaluating Limits
Evaluating Limits by Direct Substitution:
Lim (x^2 + x – 6)
x-> -1
substitute -1 in for x
(-1)^2 + (-1) – 6 = -6
Dividing out Technique:
Lim (x^2 + x – 6/x +3)
Begin by factoring the numerator and dividing out any common factors
Lim (x^2 + x – 6/x +3) = lim (x -2)(x +3)/(x + 3)
= lim (x -2)
= ((-3) – 2)
= -5
This procedure for evaluating a limit is called the dividing out technique. The validity of the procedure stems from the fact that if two functions agree at all but a single number c, they must have identical limit behavior at x = c.
Rationalizing Technique:
By direct substitution we get the indeterminate form 0/0:
In this case, we change the form of the fraction by rationalizing (eliminating the radical in) the numerator:
Therefore, by substitution, we have
Definition of Continuity
lim f(x) = lim f(x) = f(c)
lim f(x) = lim f(x) = f(c)
x->c- x->c+
Finding a Derivative:
Find the derivative of f(x) = x^2
Definition of Limits at Infinity:
If f is a function and L1 and L2 are real numbers, the statements
Lim f(x) = L1 limit as x approaches negative infinite
X->infinite
Lim f(x) = L2 limit as x approaches infinite
x-> infinite
Limits at Infinity:
If r is a positive real integer, then
Lim 1/x^r = 0
x-> infinite
Furthermore, if x^r is defined when x<0, r =" 0"> - infinite
Comparing Limits at Infinity:
Lim -2x + 3/3x^2 = 1
x->infinite
= 0
If f is a function and L1 and L2 are real numbers, the statements
Lim f(x) = L1 limit as x approaches negative infinite
X->infinite
Lim f(x) = L2 limit as x approaches infinite
x-> infinite
Limits at Infinity:
If r is a positive real integer, then
Lim 1/x^r = 0
x-> infinite
Furthermore, if x^r is defined when x<0, r =" 0"> - infinite
Comparing Limits at Infinity:
Lim -2x + 3/3x^2 = 1
x->infinite
= 0
Lim -2x^2 + 3/3x^2 = 1
x-> infinite
= -2/3
x-> infinite
= -2/3
Lim -2x^3 + 3/3x^2 = 1
x-> infinite
= DNE
x-> infinite
= DNE
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