Transcript Slide 1

An introduction to limits
Limits in calculus :
This section gives some examples of how to use algebraic techniques to compute limits . these
In cludethe terms of an infinite series, the sum of
an infinite series ,the limit of afunction, The slope of aline tangent to the graph of afunction, the
area of aregion bounded by the graphs of several functions .
Find the sum of the terms of the series mentioned above as n gets very large
the sum of the series is
so limt
Try some values for x close to 2
From the chart it appears that as x tends to 2
F(x) tends to zero so
compute the slopes of the secant lines
So as ∆x tend to 2 ero so the slope of the line tangent at x=3 close and close
to 2 for any point p close to T the slope PT is given by
Or x tends to zero p close to T so the slope of line tangent at
point T
Area The
under expression
a curve :-
in the left side is known as the
derivative of f (x) at x = 3, and is denoted by f '(3).
and we will learn many techniques for the
derivative in the following chapters
In the following chapter
Then the slope of the line tangent to the graph
of f(x) = (x-2)2 + 1 at x = 3 is 2
The area below the curve
F(x) = x2 , above the x ax is right the line x=1
and left the line x=5
Using four inscribed rectangles, each
having a base of 1 unit, their
corresponding
heights are found: f(1) = 1, f (2) = 4,
f (3) = 9, and f (4) = 16. The area
computation is shown at the right.
... area = 1*1 +1*4+1*9+1*16 =30
This area is less than the actual desired area .
Use four rectangular each having abase of 1 unit
Their corresponding height are found : f(2)=4,
f(3)=9, f(4)=16 and f(5)=25
... area = 1*4+1*9+1*16+1*25=54
This area is greater than the actual desired area .
So the actual area is greater than 30 and less than 54
The actual area of the region described in
Step 1 is greater than 30 and less than 54. If
you wanted a closer approximation of the
actual area, you would use a very large
number of rectangles, each having a base of
, where n is the number of
rectangles used. The corresponding height for
each rectangle would then be f(xi), where i
represents the 1st, or 2nd, or 3rd, or 4th
rectangle of the n rectangles used.
The sum of the areas of these
rectangles is represented by
The actual area would be found by
letting n → , so
→ 0 and then find the limt
you will compute these sorts of areas,
after learning some techniques of integration.
DEFINITION OF THE LIMIT OF A FUNCTION
let f(x) approaches L as x tends to c
. ..
if
Where
and
These means that the distance from f(x) to l can be made
small by making the distance from x to c small but not 0 .
Using
Definition to verify alimit using f(x)
Use the definition
to verify that
whenever
you need to find a connection between
If you move left and right of x = 3
just 1 unit, x would be in the
interval (2,4) so that
One sided limits :-
EX :-
Note: When you write
IT does not mean that the limit exists.
The limit actually does not exist
because f (x) increases without bound as
x approaches c.
ONE-SIDED LIMITS FOR A CONDITIONAL FUNCTION
From the left, you can see that
And From the right, you can see that
ONE-SIDED LIMITS FOR A POLYNOMIAL FUNCTION
Determine Limits from the Graph of a Function
As x approaches -1 from the left and then from
the right, two different limits so
is non exist .
From left at right
LIMITS OF A TRIGONOMETRIC FUNCTION
To the right is the graph of f(x) = sinx. Determine
each limit below.
lim (sinx) is nonexistent, sinx oscillates between –1 and 1.
Properties of Limits :Let k and c
be constants, let n be a positive
integer, and let f and g be
functions such that
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2
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3
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Special Trigonometric Limits:
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Definition of a Function Continuous at a Point
The function f is continuous at the
number c if the following conditions are
satisfied:
Another test for continuity by viewing
the function’s graph is that the graph has
no holes, no jumps, and no vertical
asymptotes.
An example of the first figure is
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Condition iii of the definition fails.Note the “hole” at x = 3. This is also
known as a removable discontinuity.