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The Limit of a function
Think of a function, say
y=x+3
OR
f(x)=x+3
As x gets close to 2, what happens to the
vaue of y or f(x)?
We can look at this three ways:
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Numerically
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Graphically
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Algebraically
Numerical: where f(x)=x+3
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Try letting x=2 ,
f(2) =5
Try numbers for x that are very close to 2
f(1.9)= ,
f(1.9999)=
,
f(2.1)= , f(2.0001)=
SO… as x gets close to 2 from below
(e.g.1.999) or from above (e.g.2.0001), y
gets close to
.
To find the limit numerically:
Try to plug in
the number.
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If you get a real number
you have won the day. Not
true for 0 .
If you get 5/0 or -2/0 then
there is no real solution to
this limit question. It is
undefined.
If you get 0/0 it is
indeterminant so you must
do more work.
Indeterminate???
To numerically find y, try numbers very
close to the given number for x. You
must try numbers BOTH from below
and from above the given value of x.
(e.g. try 1.9999 and 2.0001).
Graphically
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When you look at the graph of f(x) see
what happens to y when x is near the
number.
Be sure to look at both sides of the xvalue. They must converge to the same
point.
Algebraically
It is often easier and safer to find the
limit by an algebra trick. There are
three tricks you can watch for:
 Rationalize the numerator or
denominator.
 Get common denominators if fractional.
 Factor the numerator (or denominator).
Try some functions
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Consider f(x)=1/(x-2) as x->2
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Consider f(x)=(x2-4)/(x-2) as x->2