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Evaluating
the limit of a rational function at a point
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A
rational function is continuous at every x
except for the zeros of the denominator.
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Therefore,
all real numbers
x
except for the zeros of the denominator, is the domain of a rational function.
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The
zeros of the numerator that are in the domain are the x-intercepts
or roots of a rational function.
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If
x
= a is
a zero of the numerator and not a zero of the denominator,
then f(a)
= 0 or a
is the root of the rational
function. While if both the numerator and denominator are zero,
we get the indeterminate form 0/0.
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Factoring
the numerator and denominator into irreducible factors allows us
to find all of the zeros of the numerator
and denominator.
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a)
The limit of a rational function that is defined at the given point
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Example:
Evaluate
the limit |
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b)
The limit of a rational function that is not defined at the
given point
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At
every point that is a zero of the denominator a rational
function has either a vertical asymptote or a hole in the
graph cased by the
indeterminate form 0/0. |
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The
vertical asymptote is called the infinite discontinuity while
the hole in the graph is called removable discontinuity
since the
indeterminate form can be avoided by canceling common factors in
the numerator and the
denominator. |
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Thus,
a point of discontinuity or hole in the graph exists when a zero
of the numerator is matched by a zero of
the
denominator and the factor occurs to the same degree in the
numerator and the denominator. |
| Given a
rational function |
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then, |
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- if q
(a)
= 0 and
p
(a) is
not 0
then one-sided limits |
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are
infinite limits. |
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That means, the rational function has the vertical asymptote at x
= a. |
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- if q
(a)
= 0 and
p
(a)
= 0
the
polynomials p
(x) and
q
(x)
have a common factor (x
-
a). |
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The rational function has the removable discontinuity or the
hole in the graph at the point x
= a. |
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When
both the numerator and denominator of a rational function vanish
at the given point a,
we factor and cancel
common factors and then find the limit of the equivalent
function.
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The limit of a rational function at infinity
containing roots (irrational expressions)
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We
use the same method we used to
evaluate the limit of a rational function at infinity
that is, isolate and cancel
a common factor of x
from both the numerator and denominator and than find the limit
of the equivalent expression.
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| Example:
Evaluate
the limit |
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| Solution: |
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| Example:
Evaluate
the limit |
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| Solution: |
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The limit of a rational function at a point containing
irrational expressions, use of substitution
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Use of the method of substitution to avoid the indeterminate form of an expression.
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| Example:
Evaluate
the limit |
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Solution: Let substitute,
x
+ 1 = y6,
then as x
®
0,
y
®
1
therefore
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| Example:
Evaluate
the limit |
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Solution: Let substitute,
x
= y12,
then as x
®
1 and
y
®
1,
therefore
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Evaluating
the limit of a rational function
containing irrational expressions using rationalization
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To
avoid the
indeterminate form of the irrational expression we rationalize
the numerator or the denominator as
appropriate.
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| Example:
Evaluate
the limit |
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Solution: Let rationalize
the numerator,
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| Example:
Evaluate
the limit |
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Solution: Let rationalize
both the numerator
and denominator,
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