Functions, Limits and Continuity -3

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Session
Functions, Limits and Continuity -3
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Session Objectives

• Limit at Infinity
• Continuity at a Point
• Continuity Over an Open/Closed Interval
• Sum, Product and Quotient of Continuous Functions
• Continuity of Special Functions

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Limit at Infinity
A GEOMETRIC EXAMPLE Let's look at a polygon
inscribed in a circle... If we increase the
number of sides of the polygon, what can you say
about the polygon with respect to the circle?
As the number of sides of the polygon increase,
the polygon is getting closer and closer to
becoming the circle! If we refer to the polygon
as an n-gon, where n is the number of sides, Then
we can write
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Limit at Infinity (Cont.)
The n-gon never really gets to be the circle, but
it will get very close! So close, in fact, that,
for all practical purposes, it may as well be the
circle. That's what limits are all about!
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Limit at Infinity (Cont.)
A GRAPHICAL EXAMPLE Now, let's look at the
graph of f(x)1/x and see what happens!
Let's look at the blue arrow first. As x gets
really, really big, the graph gets closer and
closer to the x-axis which has a height of 0. So,
as x approaches infinity, f(x) is approaching 0.
This is called a limit at infinity.
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Limit at Infinity (Cont.)
Now let's look at the green arrow... What is
happening to the graph as x gets really, really
small? Yes, the graph is again getting closer and
closer to the x-axis (which is 0.) It's just 
coming in from below this time.
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Some Results
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Example - 1
Solution
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Example 2
Solution
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Example - 3
Solution
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Solution Cont.
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Example 4
Solution
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Solution (Cont.)
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Continuity at a Point
Let f(x) be a real function and let x a be any
point in its domain. Then f(x) is said to be
continuous at x a, if
If f(x) is not continuous at x a, then it is
said to be discontinuous at x a.
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Left and Right Continuity
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Continuity Over an Open/Closed Interval
f(x) is said to be continuous on (a, b) if f(x)
is continuous at every point on (a, b).
f(x) is said to be continuous on a, b if
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Sum, Product and Quotient of Continuous Functions
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Continuity of Special Functions
(1) A polynomial function is continuous
everywhere.
(2) Trigonometric functions are continuous in
their respective domains.
(4) The logarithmic function is continuous in its
domain.
(5) Inverse trigonometric functions are
continuous in their domains.
(6) The composition of two continuous functions
is a continuous function.
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Example 5
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Solution (Cont.)
So, f(x) is continuous at x 0.
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Example 6
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Solution (Cont.)
So, f(x) is continuous at x 0.
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Example 7
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Solution (Cont.)
So, f(x) is discontinuous at x 0.
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Example 8
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Solution (Cont.)
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Example 9
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Solution (Cont.)
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Example 10
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Solution (Cont.)
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