If , what value does
approach as
gets infinitely larger?
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The correct answer is E.
One way to determine the value that approaches as
gets infinitely larger is to rewrite the definition of the function to use only negative powers of
and then reason about the behavior of negative powers of
as
gets infinitely larger. Since the question is only concerned with what happens to
as
gets infinitely larger, one can assume that
is positive. For
, the expression
is equivalent to the expression
. As
gets infinitely larger, the expression
approaches the value
, so as
gets infinitely larger, the expression
approaches the value
. Thus, as
gets infinitely larger,
approaches
.
Alternatively, one can use a graphing calculator to estimate the height of the horizontal asymptote for the function . Graph the function
on an interval with “large”
, say, from
to
.
By examining the graph, the all seem very close to
. Graph the function again, from, say,
to
.
The vary even less from
. In fact, to the scale of the coordinate plane shown, the graph of the function
is nearly indistinguishable from the asymptotic line
. This suggests that as
gets infinitely larger,
approaches
, that is,
.
Note: The algebraic method is preferable, as it provides a proof that guarantees that the value approaches is
. Although the graphical method worked in this case, it does not provide a complete justification; for example, the graphical method does not ensure that the graph resembles a horizontal line for “very large”
such as
.