Limits
For a function f(x),
the limit of the function is the value that the function approaches
as x approaches a certain number. Here’s an example:
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What
is the limit of f(x) = (x +2)/(x2 –
2x – 8) as x approaches
–2? |
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Normally, finding the limit should be easy. You would
simply plug the value that x approaches, in this
case –2, into the function and produce the limit. But the Math IC
typically asks only one type of question about limits—the limit
of a function at a point at which the function is not defined. Try
and plug –2 into the function.
It seems that the function is not defined at x =
–2, because division by zero is not allowed. This is precisely what
the test wants you to think. But the assumption
that this function is undefined at –2 is incorrect. Luckily, there
is an easy way to solve for the limit.
First, you need to factor the function so that it is in
its most simplified state. For our example, the denominator can
be factored:
Once the denominator has been factored, it’s easy to see
that the function x+2⁄(x+2)(x–4) simplifies
to 1⁄x–4.
This simplified fraction can be evaluated at x =
–2:
1⁄–6 is
the limit of f(x) at x =
–2.
You should choose “the limit does not exist at this point”
answer only if a function is undefined at the point at which you
wish to find a limit and the function cannot be factored any further.
=x+2.gif)
=1-6.gif)
