the right. But when we have , the function’s sign depends on the sign of x, and you get a different limit
from each side.
Let’s look at a few examples in which the independent variable approaches infinity.
(1)If the left-hand limit of a function is not equal to the right-hand limit of the function, then the
limit does not exist.
(2)A limit equal to infinity is not the same as a limit that does not exist, but sometimes you will
see the expression “no limit,” which serves both purposes. If = ∞, the limit,
technically, does not exist.(3)If k is a positive constant, then = ∞, = −∞, and does not exist.(4)If k is a positive constant, then , and .Example 6: Find .
As x gets bigger and bigger, the value of the function gets smaller and smaller. Therefore, = 0.
Example 7: Find .
It’s the same situation as the one in Example 6; as x decreases (approaches negative infinity), the value of
the function increases (approaches zero). We write the following:
= 0
We don’t have the same problem here that we did when x approached zero because “positive zero” is the
same thing as “negative zero,” whereas positive infinity is different from negative infinity.
Here’s another rule.
If k and n are constants, |x| > 1, and n > 0, then = 0, and = 0.