Limits and Continuity

1
Limits and Continuity

• Theorems and Examples

2
Sandwich Theorem

• Suppose h(x) lt f(x) lt g(x) for all x.

g (x)
h(x)
3
Sandwich Theorem
0
0
g (x)

h(x)
4
Sandwich Theorem
0
g (x)
h(x)
5
Continuous Functions

• Definition A function, f(x) is continuous at a
  if

and
f(a) L
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Properties of Continuous Functions

• Suppose f(x) and g(x) are continuous, then
• f(x) g(x)
• f(x) g(x)
• f(x)g(x)
• f(x)/g(x) , g(x) ?0
• f(g(x))
• Are all continuous

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If f(x) and g(x) are continuous, does that mean f
g(x) is continuous?

• PROOF

By definition
By definition
By the addition property of limit