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## Limits

Properties of Limits

If b and c are real numbers, n is a positive integer, and the functions ƒ and g have limits as , then the following properties are true.

 1. Scalar multiple: [b(ƒ(x))] = b[ ƒ(x)] 2. Sum or difference: [ƒ(x) g(x)] = ƒ(x)  g(x) 3. Product: [ƒ(x)g(x)] = [ ƒ(x)][ g(x)] 4. Quotient: [ƒ(x)/g(x)] = [ ƒ(x)]/[ g(x)], if g(x) 0

One-Sided Limits ƒ(x) x approaches c from the right ƒ(x) x approaches c from the left

Limits at Infinity ƒ(x) = L or ƒ(x) = L

The value of ƒ(x) approaches L as x increases/decreases without bound.

y = L is the horizontal asymptote of the graph of ƒ.

Some Nonexistent Limits     sin Some Infinite Limits    ln x Exercise: What is  ? (A) 1 (B) 0 (C) (D) (E) The limit does not exist. /TD> The answer is A. You should memorize this limit.

Continuity

Definition

A function ƒ is continuous at c if:

1. ƒ(c) is defined

2. ƒ(x) exists

3. ƒ(x) = ƒ(c)

Graphically, the function is continuous at c if a pencil can be moved along the graph of ƒ(x) through (c, ƒ(c)) without lifting it off the graph.

 Exercise: If for x 0 and if ƒ is continuous at x = 0, then k = (A) -3/2 (B) -1 (C) 0 (D) 1 (E) 3/2 The answer is E. ƒ(x) = 3/2

Intermediate Value Theorem

If ƒ is continuous on [a, b] and k is any number between ƒ(a) and ƒ(b), then there is at least one number c between a and b such that ƒ(c) = k.    Next: Differential Calculus Up: Topics to Study Previous: Elementary Functions