Introduction to Limits

Limits are the foundation of calculus. They describe what happens to a function as its input approaches a particular value. Even though you may not be able to plug in that value directly, limits tell you what the function "wants to be." This concept opens the door to derivatives and integrals.

The Concept of a Limit

The limit of f(x) as x approaches a value c is written as:

lim (x-->c) f(x) = L

This means: as x gets closer and closer to c (but not equal to c), f(x) gets closer and closer to L.

Example: lim (x-->2) (x² - 4)/(x - 2)

If you try x = 2, you get 0/0 (undefined). But simplifying:

• (x² - 4)/(x - 2) = (x+2)(x-2)/(x-2) = x + 2 (for x not equal to 2)
• lim (x-->2) (x + 2) = 4

Limit Laws

If lim (x-->c) f(x) = L and lim (x-->c) g(x) = M, then:

1. Sum rule: lim [f(x) + g(x)] = L + M
2. Difference rule: lim [f(x) - g(x)] = L - M
3. Product rule: lim [f(x) . g(x)] = L . M
4. Quotient rule: lim [f(x)/g(x)] = L/M (if M is not 0)
5. Power rule: lim [f(x)]ⁿ = Lⁿ
6. Constant multiple: lim [k . f(x)] = k . L

Indeterminate Forms

When direct substitution gives forms like 0/0, inf/inf, 0 x inf, etc., the limit is called indeterminate. We need algebraic manipulation to resolve it.

Common techniques:

• Factoring: Cancel common factors
• Rationalizing: Multiply by conjugate (useful with square roots)
• Division: Divide numerator and denominator by the highest power of x

Worked Examples

Example 1 (Factoring): lim (x-->3) (x² - 9)/(x - 3) = lim (x-->3) (x+3)(x-3)/(x-3) = lim (x-->3) (x+3) = 6

Example 2 (Rationalizing): lim (x-->0) (sqrt(x+1) - 1)/x

Multiply by conjugate: (sqrt(x+1) - 1)(sqrt(x+1) + 1) / [x(sqrt(x+1) + 1)] = (x+1-1) / [x(sqrt(x+1) + 1)] = x / [x(sqrt(x+1) + 1)] = 1 / (sqrt(x+1) + 1)

As x-->0: 1/(sqrt(1) + 1) = 1/2 = 0.5

Example 3 (Limits at infinity): lim (x-->inf) (3x² + 2)/(x² - 1)

Divide by x²: lim (x-->inf) (3 + 2/x²)/(1 - 1/x²) = 3/1 = 3

Important Standard Limits

lim (x-->0) sin(x)/x = 1 (x in radians)

lim (x-->0) (1 - cos(x))/x = 0

lim (x-->inf) (1 + 1/x)ˣ = e (where e = 2.718...)

Continuity

A function f(x) is continuous at x = c if:

1. f(c) is defined
2. lim (x-->c) f(x) exists
3. lim (x-->c) f(x) = f(c)

If any condition fails, the function is discontinuous at that point.

Key Takeaways

• Limits describe function behaviour as input approaches a value
• Indeterminate forms (0/0) require algebraic techniques to resolve
• Standard limits like sin(x)/x = 1 are frequently used
• Continuity means the function equals its limit at every point