Chapter 1.3
Evaluating Limits Analytically
Direct Substitution can be used to evaluate a limit at c -- if the function is continuous at c
Theorem 1.1 - Some Basic Limits:
Let b and c be real numbers and let n be a positive integer
lim (-^- (x -> c)) b = b
lim (-^- (x -> c)) x = c
lim (-^- (x -> c)) x^n = c^n
Theorem 1.2 - Properties of Limits:
Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:
lim (-^- (x -> c)) f(x) = L
lim (-^- (x -> c)) g(x) = K
1. Scalar Multiple:
lim (-^- (x -> c)) [b f(x)] = bL
2. Sum or Difference:
lim (-^- (x -> c)) [f(x) ± g(x)]
3. Product:
lim (-^- (x -> c)) [f(x) g(x)] = LK
4. Quotient:
lim (-^- (x -> c)) [f(x)/g(x)] = L/K
- provided that K ≠ 0
5. Power:
lim (-^- (x -> c)) [f(x)]^n = L^n
Theorem 1.3 - Limits of Polynomial and Rational Functions:
If p is a polynomial function and c is a real number, then
lim (-^- (x -> c)) p(x) = p(c)
If r is a rational function given by r(x) = p(x) / q(x) and c is a real number such that q(c) ≠ 0, then
lim (-^- (x -> c)) r(x)
= r(c) = p(c) / q(c)
Theorem 1.4 - The Limit of a Function Involving a Radical:
Let n be a positive integer. The following limit is valid for all c if n is odd, and it is valid for c > 0 if n is even:
lim (-^- (x -> c)) n√(x)
= n√(c)
Theorem 1.5 - The Limit of a Composite Function:
If f and g are functions such that
lim (-^- (x -> c)) g(x) = L
and
lim (-^- (x -> c)) f(x) = f(L)
then
lim (-^- (x -> c)) f(g(x))
= f (lim (-^- (x -> c)) g(x) )
= f(L)
Theorem 1.6 - Limits of Trigonometric Functions:
Let c be a real number in the domain of the given trig function
1. lim (-^- (x -> c)) sin x = sin c
2. lim (-^- (x -> c)) cos x = cos c
3. lim (-^- (x -> c)) tan x = tan c
4. lim (-^- (x -> c)) sec x = sec c
5. lim (-^- (x -> c)) csc x = csc c
6. lim (-^- (x -> c)) cot x = cot c
Theorem 1.7 - Functions That Agree at All but One Point:
Let c be a real number and let f(x) = g(x) for all x ≠ c in an open interval containing c. If the limit of g(x) as x approaches c exists, then the limit of f(x) also exists and
lim (-^- (x -> c)) f(x)
= lim (-^- (x -> c)) g(x)
A Strategy for Finding Limits
1. Recognize which limits can be evaluated by direct substitution (Theorems 1.1 - 1.6)
2. If the limit can't be evaluated with direct substitution, find a function g that agrees with f for all x other than x = c
(Make sure you use a g where the limit of g(x) CAN be evaluated using direct substitution!)
3. Apply Theorem 1.7 to conclude analytically that
lim (-^- (x -> c)) f(x)
= lim (-^- (x -> c)) g(x)
= g(c)
4. Reinforce your conclusion with a graph or with a table
Dividing Out Technique:
When dealing with a rational function, factor both the numerator and the denominator completely and simplify. This will eliminate the values where there is a hole in the graph to make finding the limit easier. This will also reduce encounters with the indeterminate form.
Indeterminate Form - 0/0 - you cannot determinate the limit of the function from this form alone! So rewrite the fraction to avoid the indeterminate form!
Rationalizing Technique:
When you have a rational function with something under a radical plus (or minus) another number that is not under a radical for its numerator or denominator, you can rewrite the fraction to enhance the limit solving process. Rewrite the fraction by multiplying both the numerator and the denominator by the same radical minus (or plus) the same number not under a radical.
Theorem 1.8 - The Squeeze Theorem:
If h(x) ≤ f(x) ≤ g(x) for all x in an open interval containing c, except possibly at c itself, and if
lim (-^- (x -> c)) h(x)
= lim (-^- (x -> c)) g(x)
= L
then lim (-^- (x -> c)) f(x) exists and is equal to L.
Theorem 1.9 - Two Special Trigonometric Limits:
lim (-^- (x -> 0)) (sin x /x) = 1
lim (-^- (x -> 0)) ((1 - cos x)/x) = 0
Chapter 1.2
What Is a Limit?
Simple Definition of a Limit:
As x approaches c, the limit L is the number f(x) approaches from both the left and right sides.
(Except when we are dealing with one-sided limits, in which x approaches c only from one side.)
lim (^- (x -> c)) = L
...so how do we find a limit?
Three-Way Approach for Problem Solving:
1. Numerical Approach -- construct a table of values
2. Graphical Approach -- draw a graph
3. Analytic Approach -- use algebra or calculus
A limit doesn't always exist and there are ways for you to spot the nonexistence of a limit...
Common Types of Behavior Associated With Nonexistent Limits:
1. f(x) approaches different numbers from the left and right sides of c.
Example: Piecewise functions that break off at x = c
2. f(x) increases or decreases without bound as x approaches c.
Example: There is a vertical asymptote at c
3. f(x) oscillates between two fixed values as x approaches c.
Example: "Weird function that doesn't make sense when you zoom in and just looks like crazy patterns of zigzags"
ε - δ* Definition of a Limit:
Let f be a function defined on an open interval containing c with the exception of possibly at c itself and let L be a real number. The limit lim (^- (x -> c)) = L mean that for each ε > 0 there exists a δ > 0 such that if
0 < |x - c| < δ
(meaning three things:
1. The distance between x and c is more than 0
2. x lies either in the interval of (c - δ, c) or (c, c + δ )
3. x is within δ units of c)
then
|f(x) - L| < ε
(meaning f(x) lies in the interval of (L - ε, L + ε) )
*Just in case you were curious, ε is the Greek letter epsilon (in lowercase), and it represents a positive number, a small positive number. δ, on the other hand, is the Greek letter delta (also in lowercase).
The Dirichlet Function:
f(x) = {0, if x is rational
{1, if x is irrational
Features:
-
No limits exist for any real number c
-
Not continuous at any real number c
I know, this is a strange function... ;)
Chapter 1 and Chapter 3.5
LIMITS
Limits are the first topic covered in Calculus, and it is important to master techniques of finding different types of them.
...some music to help you relax and concentrate :)
Chapter 1.2
What Is a Limit?
Simple Definition of a Limit:
As x approaches c, the limit L is the number f(x) approaches from both the left and right sides.
(Except when we are dealing with one-sided limits, in which x approaches c only from one side.)
lim (-^- (x -> c)) = L
...so how do we find a limit?
Three-Way Approach for Problem Solving:
-
Numerical Approach -- construct a table of values
-
Graphical Approach -- draw a graph
-
Analytic Approach -- use algebra or calculus
A limit doesn't always exist and there are ways for you to spot the nonexistence of a limit...
Common Types of Behavior Associated With Nonexistent Limits:
1. f(x) approaches different numbers from the left and right sides of c.
Example: Piecewise functions that break off at x = c
2. f(x) increases or decreases without bound as x approaches c.
Example: There is a vertical asymptote at c
3. f(x) oscillates between two fixed values as x approaches c.
Example: "Weird function that doesn't make sense when you zoom in and just looks like crazy patterns of zigzags"
ε - δ* Definition of a Limit:
Let f be a function defined on an open interval containing c with the exception of possibly at c itself and let L be a real number. The limit lim (-^- (x -> c)) = L mean that for each ε > 0 there exists a δ > 0 such that if
0 < |x - c| < δ
(This means three things:
-
The distance between x and c is more than 0
-
x lies either in the interval of (c - δ, c) or (c, c + δ )
-
x is within δ units of c)
then
|f(x) - L| < ε
(meaning f(x) lies in the interval of (L - ε, L + ε) )
*Just in case you were curious, ε is the Greek letter epsilon (in lowercase), and it represents a positive number, a small positive number. δ, on the other hand, is the Greek letter delta (also in lowercase).
The Dirichlet Function:
f(x) = {0, if x is rational
{1, if x is irrational
Features:
-
No limits exist for any real number c
-
Not continuous at any real number c
I know, this is a strange function... ;)
Since it is very difficult to type on a computer to produce the limit notation, on this website I used an alternative :
lim (-^- (whatever value that x is approaching)) (whatever we're finding the limit of) = (whatever the limit is)
In future cases, keep in mind the following:
* -> multiply
/ -> divide
^ -> superscript
-^ -> subscript
n√(x) -> the nth root of x
( ) -> paranthese used for grouping to convey a more clear meaning
(I'm sure you are familiar with many of these since you probably use them very often on your scientific calculator 😉)
Chapter 1.4
Continuity and One-Sided Limits
Definition of Continuity:
Continuity at a Point - A function f is continuous at c id the following three conditions are met:
1. f(c) is defined
(vs. undefined)
2. lim (-^- (x -> c)) f(x) exists
(vs. the limit of f(x) does not exist at x = c)
3. lim (-^- (x -> c)) f(x) = f(c)
(vs. the limit of f(x) exists at x = c, but it is not equal to f(c).)
Continuity on an Open Interval - a function is continuous on an open interval (a, b) if it is continuous at each point in the interval. A function that i scontinuous on the entire real line (-∞, +∞) is everywhere continuous.
Discontinuities
We have a discontinuity when the function is not continuous.
Removable Discontinuity - can be made continuous by defining or redefining f(c)
Nonremovable Discontinuity - changing the value of f(c) does not make the function continuous
One-Sided Limits
Limit from the Right:
lim (-^- (x -> c^+)) = L
Limit from the Left:
lim (-^- (x -> c^-)) = L
One sided limits are useful in taking limits of functions involving radicals:
lim (-^- (x -> 0^+)) n√(x) = 0
Theorem 1.10 - The Existence of a Limit:
Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L if and only if
lim (-^- (x -> c^+)) = L
and
lim (-^- (x -> c^-)) = L
Definition of Continuity on a Closed Interval:
A function is continuous on the closed interval [a, b] if it is continous on the open interval (a, b) and
lim (-^- (x -> a^+)) = f(a)
lim (-^- (x -> b^-)) = f(b)
The function f is continous from the right at a and from the left at b.
Theorem 1.11 - Properties of Continuity
If b is a real number and f and g are continuous at x = c, then the following functions are also continuous at c:
-
Scalar Multiple: bf
-
Sum and Difference: f ± g
-
Product: fg
-
Quotient: f /g , if g(c) ≠ 0
Functions Continuous at Every Point in Their Domains:
-
Polynomial Functions
-
Rational Functions (when the denominator does not equal 0)
-
Radical Functions
-
Trigonometric Functions
Theorem 1.11 + This Summary = a wide variety of elementary functions are continuous at every point in their domains!
Theorem 1.12 - Continuity of a Composite Function:
If g is continuous at c and f is continuous at g(c), then the composite function given by (f ⚬g) (x) = f(g(x)) is continuous at c.
If f and g satisfies the given conditions, then the limit of f(g(x)) as x approaches c is:
lim (-^- (x -> c)) f(g(x)) = f(g(c))
Theorem 1.13 - Intermediate Value Theorem:
If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(x) = k
NOTE: This theorem tells you that at least one c exist, but it does not give you a method for finding c. Such theorems are called existence theorems.
- If f is continuous on [a,b] and f(a) and f(b) differ in sign, the Intermediate Value Theorem guarantees the existence of at least one solution of f in the closed interval [a, b].
Bisection Method for Approximating Real Zeros of a Continuous Function:
1. Since a zero exists in the closed interval [a, b], it must lie in the interval [a, (a + b) / 2] or [ (a + b) / 2, b]
2. From the sign of f( [a + b] / 2), you can determine which interval contain the zero.
3. By repeatedly bisecting the interval, you can "close in" on the zero of the function.
Chapter 1.5
Infinite Limits
Infinite Limit - a limit in which f(x) increases or decreases without bound as x approaches c
Definition of Infinite Limits:
Let f be a function that is defined at every real number in some open interval containing c (except possibly at c itself). The statement
lim (-^- (x -> c)) f(x) = ∞
means that for each M > 0 there exists a δ > 0 such that f(x) > M whenever 0 < |x - c| < δ
Similarly, the statement
lim (-^- (x -> c)) f(x) = -∞
mean that for each N < 0 there exists a δ > 0 such that f(x) < N whenever 0 < |x - c| < δ.
To define the infinite limit from the left, replace 0 < |x - c| < δ by c - δ < x < c
To define the infinite limit from the right, replace 0 < |x - c| < δ by c < x < c + δ
Definition of Vertical Asymptotes:
If f(x) approaches infinity or negative infinity as x approaches c from the right or left, then the line x = c is a vertical asymptote of the graph of f.
Theorem 1.14 - Vertical Asymptotes:
Let f and g be continuous on an open interval containing c. If f(c) ≠ 0, g(c) = 0, and there exists an open interval containing c such that g(x) ≠ 0 for all x ≠ c in the interval, then the graph of the function given by
h(x) = f(x) /g(x)
has a vertical asymptote at x = c.
Theorem 1.15 - Properties of Infinite Limits:
Let c and L be real numbers and let f and g be functions such that
lim (-^- (x -> c)) f(x) = ∞
and
lim (-^- (x -> c)) g(x) = L
1. Sum or difference:
lim (-^- (x -> c)) [f(x) ± g(x)] = ∞
2. Product:
lim (-^- (x -> c)) [f(x) g(x)] = ∞, L > 0
lim (-^- (x -> c)) [f(x) g(x)] = -∞, L < 0
3. Quotient:
lim (-^- (x -> c)) [f(x) /g(x)] = 0
Similar properties hold for one-sided limits and for functions for which the limit of f(x) as x approaches c is -∞.
Chapter 3.5
Limits at Infinity
Limits at Infinity: the value that f(x) approaches as x increases or decreases without bound.
Definition of Limits at Infinity:
Let L be a real number.
1. The statement lim (-^- (x -> ∞)) f(x) = L mean:
for each 𝈡 > 0
there exists an M > 0
such that |f(x) - L| < 𝈡
whenever x > M.
(f(x) is within 𝈡 units of L as x -> ∞)
2. The statement lim (-^- (x -> -∞)) f(x) = L means:
for each 𝈡 > 0
there exists an N < 0
such that |f(x) - L| < 𝈡
whenever x < N.
Definition of Horizontal Asymptote:
The line y = L is a horizontal asymptote of the graph of f if
lim (-^- (x -> ∞)) f(x) = L
or if
lim (-^- (x -> -∞)) f(x) = L
Theorem 3.10 - Limits at Infinity:
If r is a positive rational number and c is any real number, then
lim (-^- (x -> ∞)) (c /x^r) = 0
Furthermore, if x^r is defined when x < 0, then
lim (-^- (x -> -∞)) (c /x^r) = 0
Indeterminate Form: ∞/∞
Finding Limits at ±∞ of Rational Functions:
BOBO BOTN EATS DC
BOBO - If the degree is bigger on bottom, then y = 0
BOTN - If the degree is bigger on top, then there is no limit (in other words, the limit does not exist)
EATS DC - If the exponents are the same, divide coefficients.
Definition of Infinite Limits at Infinity:
Let f be a function defined on the interval (a, ∞)
1. The statement lim (-^- (x -> ∞)) f(x) = ∞ means:
for each positive number M,
there is a corresponding number N > 0
such that f(x) > M whenever x > N.
2. The statement lim (-^- (x -> ∞)) f(x) = -∞ means:
for each negative number M,
there is a corresponding number N > 0
such that f(x) < M whenever x > N
A Tip for Finding the Limit for Rational Functions with 2 Horizontal Asymptotes:
- Divide both the numerator and the denominator by the variable used raised to the highest power present in the entire rational function.
- When radicals are present,
write x as √(x^2) for x > 0
write x as -√(x^2) for x < 0
- Take the limit.
