Introduction to Limits

What is a Limit?

Limit: Value that a function approaches as input approaches some value

Notation: lim(x→a) f(x) = L

Read as: "The limit of f(x) as x approaches a equals L"

Meaning: As x gets closer to a, f(x) gets closer to L

Important: Limit describes behavior near a, not necessarily at a

Intuitive Understanding

Imagine walking toward a point:

• Limit = where you're heading
• You might never reach it
• Function might not even be defined there

Example 1: Simple Function

f(x) = 2x + 1

Find: lim(x→3) f(x)

As x approaches 3:

• x = 2.9 → f(x) = 6.8
• x = 2.99 → f(x) = 6.98
• x = 2.999 → f(x) = 6.998

Approaching 7 from below

• x = 3.1 → f(x) = 7.2
• x = 3.01 → f(x) = 7.02
• x = 3.001 → f(x) = 7.002

Approaching 7 from above

lim(x→3) (2x + 1) = 7

Example 2: Evaluating at the Point

For continuous functions:

lim(x→a) f(x) = f(a)

Just substitute!

lim(x→2) (x² + 3x) = 2² + 3(2) = 10

When Limits Don't Equal Function Value

Limit can exist even if f(a) is undefined

Example: Hole in Graph

f(x) = (x² - 4)/(x - 2)

Find: lim(x→2) f(x)

Direct substitution: 0/0 (indeterminate)

Simplify first:

f(x) = (x² - 4)/(x - 2)
     = (x - 2)(x + 2)/(x - 2)
     = x + 2  (for x ≠ 2)

Now find limit:

lim(x→2) (x + 2) = 4

Answer: 4 (even though f(2) undefined!)

Graph has hole at (2, 4)

One-Sided Limits

Left-hand limit: lim(x→a⁻) f(x)

• Approach from left (values < a)

Right-hand limit: lim(x→a⁺) f(x)

• Approach from right (values > a)

Two-sided limit exists if: lim(x→a⁻) f(x) = lim(x→a⁺) f(x)

Example: Piecewise Function

f(x) = x + 1 for x < 2 f(x) = 2x for x ≥ 2

Find: lim(x→2) f(x)

From left (x < 2):

lim(x→2⁻) f(x) = lim(x→2⁻) (x + 1) = 3

From right (x ≥ 2):

lim(x→2⁺) f(x) = lim(x→2⁺) (2x) = 4

Since 3 ≠ 4: lim(x→2) f(x) does not exist

Basic Limit Properties

If lim(x→a) f(x) = L and lim(x→a) g(x) = M:

Sum: lim [f(x) + g(x)] = L + M

Difference: lim [f(x) - g(x)] = L - M

Product: lim [f(x) · g(x)] = L · M

Quotient: lim [f(x) / g(x)] = L/M (if M ≠ 0)

Constant multiple: lim [k · f(x)] = k · L

Example: Using Properties

lim(x→3) (2x² + 5x - 1)

= 2·lim(x→3) x² + 5·lim(x→3) x - lim(x→3) 1

= 2(9) + 5(3) - 1

= 18 + 15 - 1 = 32

Limits at Infinity

Behavior as x gets very large

lim(x→∞) f(x): As x → ∞

lim(x→-∞) f(x): As x → -∞

Example 1: Polynomial

lim(x→∞) (3x² + 2x - 5)

As x → ∞, highest power dominates

= lim(x→∞) 3x² = ∞

Example 2: Rational Function

lim(x→∞) (2x + 1)/(x + 3)

Divide numerator and denominator by highest power (x):

= lim(x→∞) (2 + 1/x)/(1 + 3/x)

As x → ∞, 1/x → 0:

= (2 + 0)/(1 + 0) = 2

Answer: 2 (horizontal asymptote)

Example 3: Different Degrees

lim(x→∞) x/(x² + 1)

Degree of denominator > numerator

Divide by x²:

= lim(x→∞) (1/x)/(1 + 1/x²)
= 0/1 = 0

Infinite Limits

Function approaches ±∞

lim(x→a) f(x) = ∞: Vertical asymptote

Example: Vertical Asymptote

lim(x→2) 1/(x - 2)²

As x → 2, denominator → 0

Numerator stays 1

= ∞ (vertical asymptote at x = 2)

Indeterminate Forms

Forms that need algebraic manipulation:

0/0: Factor and cancel

∞/∞: Divide by highest power

∞ - ∞, 0·∞, 1^∞: Various techniques

Example: 0/0 Form

lim(x→1) (x² - 1)/(x - 1)

Direct substitution: 0/0

Factor:

= lim(x→1) (x - 1)(x + 1)/(x - 1)
= lim(x→1) (x + 1)
= 2

Continuity

Function is continuous at a if:

1. f(a) is defined
2. lim(x→a) f(x) exists
3. lim(x→a) f(x) = f(a)

All three conditions must hold!

Example 1: Continuous

f(x) = x² at x = 3

1. f(3) = 9 ✓
2. lim(x→3) x² = 9 ✓
3. Limit = value ✓

Continuous at x = 3

Example 2: Not Continuous

f(x) = (x² - 9)/(x - 3) at x = 3

1. f(3) = 0/0 undefined ✗

Not continuous (has removable discontinuity)

Types of Discontinuities

Removable: Hole (can be "fixed")

Jump: Different one-sided limits

Infinite: Vertical asymptote

Example: Jump Discontinuity

f(x) = x for x < 0 f(x) = x + 2 for x ≥ 0

At x = 0:

• lim(x→0⁻) = 0
• lim(x→0⁺) = 2
• Jump discontinuity

Real-World Interpretation

Instantaneous rate of change (leads to derivative)

Approaching a boundary (physical limits)

Asymptotic behavior (long-term trends)

Example: Velocity

Position s(t) = 16t²

Average velocity from t=1 to t=2:

[s(2) - s(1)]/(2 - 1) = [64 - 16]/1 = 48

Instantaneous velocity at t=1:

lim(h→0) [s(1+h) - s(1)]/h
= lim(h→0) [16(1+h)² - 16]/h
= lim(h→0) [16 + 32h + 16h² - 16]/h
= lim(h→0) [32h + 16h²]/h
= lim(h→0) (32 + 16h)
= 32

This is the derivative! (calculus)

Bridge to Calculus

Limits are foundation of calculus:

Derivative: lim(h→0) [f(x+h) - f(x)]/h

Integral: Limit of Riemann sums

Continuity: Essential for many theorems

Practice