Comprehensive Notes on Limits
Table of Contents
1. Definition of Limits
A limit describes the value a function approaches as the input approaches a particular value.
Formal Definition:
The limit of a function f(x) as x approaches a value c is L, written as:
limx→c f(x) = L
If for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε.
In simpler terms, the limit L is the value that f(x) gets arbitrarily close to as x gets arbitrarily close to c, from either direction.
Important Note:
The limit of a function at a point does not depend on the value of the function at that point. The function may not even be defined at that point.
2. Limit Notation
Limit notation uses the following format:
- limx→c f(x) = L: The limit of f(x) as x approaches c is L
- limx→c+ f(x) = L: The right-hand limit (as x approaches c from values greater than c)
- limx→c- f(x) = L: The left-hand limit (as x approaches c from values less than c)
- limx→∞ f(x) = L: The limit as x approaches infinity
- limx→-∞ f(x) = L: The limit as x approaches negative infinity
For a limit to exist:
The left-hand limit must equal the right-hand limit:
limx→c- f(x) = limx→c+ f(x) = L
3. Types of Limits
3.1 Two-sided Limits
These are the standard limits where x approaches a value c from both directions.
Example:
limx→2 (x² - 4) = limx→2 (x - 2)(x + 2) = 0 · 4 = 0
3.2 One-sided Limits
These limits approach a value from only one direction (left or right).
Example:
For f(x) = √(x - 1):
limx→1+ √(x - 1) = 0 (right-hand limit exists)
limx→1- √(x - 1) does not exist (cannot take square root of negative numbers)
3.3 Limits at Infinity
These describe the behavior of a function as x grows arbitrarily large (or negatively large).
Example:
limx→∞ (3x² + 2x - 5)/(x² + 1) = limx→∞ (3 + 2/x - 5/x²)/(1 + 1/x²) = 3/1 = 3
3.4 Infinite Limits
These are limits where the function values grow without bound as x approaches a value.
Example:
limx→0 1/x² = ∞ (approaches infinity from both sides)
limx→0- 1/x = -∞ (approaches negative infinity from the left)
limx→0+ 1/x = ∞ (approaches positive infinity from the right)
4. Methods for Solving Limits
4.1 Direct Substitution
If the function is continuous at x = c, simply substitute the value:
Example:
limx→3 (x² + 5x - 2) = 3² + 5(3) - 2 = 9 + 15 - 2 = 22
4.2 Factoring
Use factoring when you have an expression with a removable discontinuity:
Example:
limx→-3 (x² - 9)/(x + 3) = limx→-3 (x - 3)(x + 3)/(x + 3) = limx→-3 (x - 3) = -3 - 3 = -6
4.3 Rationalization
Multiply numerator and denominator by the conjugate when dealing with square roots:
Example:
limx→4 (√x - 2)/(x - 4) = limx→4 [(√x - 2)(√x + 2)]/[(x - 4)(√x + 2)] = limx→4 (x - 4)/[(x - 4)(√x + 2)] = limx→4 1/(√x + 2) = 1/4
4.4 Using Trigonometric Identities
Apply trigonometric identities to simplify expressions with trigonometric functions:
Example:
limx→0 (sin 2x)/(sin x) = limx→0 (2 sin x cos x)/(sin x) = limx→0 2 cos x = 2 cos(0) = 2
4.5 Dividing by Highest Power
For limits at infinity, divide both numerator and denominator by the highest power of x:
Example:
limx→∞ (2x³ - 3x² + 4)/(5x³ + x) = limx→∞ [(2 - 3/x + 4/x³)/(5 + 1/x²)] = 2/5
4.6 L'Hôpital's Rule
If a limit gives an indeterminate form (0/0 or ∞/∞), you can differentiate the numerator and denominator separately:
Example:
limx→0 (sin x)/x = 0/0 indeterminate form
Apply L'Hôpital's rule: limx→0 (sin x)/x = limx→0 (cos x)/1 = cos(0) = 1
4.7 Using Squeeze Theorem
If g(x) ≤ f(x) ≤ h(x) for all x near c (except possibly at c) and limx→c g(x) = limx→c h(x) = L, then limx→c f(x) = L.
Example:
limx→0 x² sin(1/x)
Since -1 ≤ sin(1/x) ≤ 1 for all x ≠ 0, we have -x² ≤ x² sin(1/x) ≤ x²
As x→0, both -x² and x² approach 0, so by the squeeze theorem, limx→0 x² sin(1/x) = 0
5. Special Limits
These important limits form the foundation for calculus:
-
limx→0 (sin x)/x = 1
-
limx→0 (1 - cos x)/x = 0
-
limx→0 (tan x)/x = 1
-
limx→0 (ex - 1)/x = 1
-
limx→0 (1 + x)1/x = e
-
limx→∞ (1 + 1/x)x = e
-
limx→0 ln(1 + x)/x = 1
6. Indeterminate Forms
Indeterminate forms are expressions that don't give a clear limit value and require special techniques:
0/0 Form
Example: limx→0 (sin x)/x
Solution: Use L'Hôpital's rule to get limx→0 (cos x)/1 = 1
∞/∞ Form
Example: limx→∞ (2x²+3x)/(x²-1)
Solution: Divide top and bottom by highest power x² to get limx→∞ (2+3/x)/(1-1/x²) = 2/1 = 2
0·∞ Form
Example: limx→0+ x·ln(x)
Solution: Rewrite as limx→0+ ln(x)/1/x which gives the ∞/∞ form. Apply L'Hôpital's rule to get limx→0+ 1/x · (-x²) = limx→0+ -x = 0
∞-∞ Form
Example: limx→∞ (√(x²+x) - x)
Solution: Rationalize to get limx→∞ (x²+x-x²)/(√(x²+x)+x) = limx→∞ x/(√(x²+x)+x) = limx→∞ 1/(√(1+1/x)+1) = 1/2
1∞ Form
Example: limx→∞ (1+2/x)x
Solution: Use the property limx→∞ (1+a/x)x = ea to get e2
00 Form
Example: limx→0+ xx
Solution: Let y = xx, then ln(y) = x·ln(x). As x→0+, x·ln(x)→0, so ln(y)→0 which means y→1
∞0 Form
Example: limx→∞ x1/ln(x)
Solution: Let y = x1/ln(x), then ln(y) = ln(x)/ln(x) = 1, so y = e
7. Applications of Limits
Derivatives
The derivative of a function f(x) at x = a is defined as:
f'(a) = limh→0 [f(a+h) - f(a)]/h
Example: Find the derivative of f(x) = x² at x = 3
f'(3) = limh→0 [(3+h)² - 3²]/h = limh→0 [9+6h+h² - 9]/h = limh→0 [6h+h²]/h = limh→0 [6+h] = 6
Continuity
A function f(x) is continuous at x = a if and only if:
limx→a f(x) = f(a)
Example: Is f(x) = (x² - 4)/(x - 2) continuous at x = 2?
limx→2 (x² - 4)/(x - 2) = limx→2 (x - 2)(x + 2)/(x - 2) = limx→2 (x + 2) = 4
But f(2) is undefined, so f(x) is not continuous at x = 2.
Rate of Change
Limits help determine instantaneous rates of change:
Example: If s(t) = 16t² represents the position of an object (in feet) at time t (in seconds), find the instantaneous velocity at t = 3.
v(3) = limh→0 [s(3+h) - s(3)]/h = limh→0 [16(3+h)² - 16(3)²]/h
= limh→0 [16(9+6h+h²) - 16(9)]/h = limh→0 [16(6h+h²)]/h = limh→0 16(6+h) = 96 ft/s
Area Under a Curve
Limits form the basis for finding the area under a curve through integration:
A = limn→∞ Σi=1n f(ci)·Δx
Where Δx = (b-a)/n and ci is a point in the i-th subinterval.
8. Quiz on Limits
Test your understanding of limits with this quiz
Select your answer and click "Check Answer" to see if you're correct.