power of x in the expression, which is x^4 : = . Next,
simplify the top and bottom: . Now, if we take the limit as x goes to infinity,
we get = .
6. + ∞
Here we have to think about what happens when we plug in a value that is very close to 6, but
a little bit more. The top expression will approach 8. The bottom expression will approach 0,
but will be a little bit bigger. Thus, the limit will be , which is +∞.
7. −∞
Here we have to think about what happens when we plug in a value that is very close to 6, but
a little bit less. The top expression will approach 8. The bottom expression will approach 0
but will be a little bit less. Thus, the limit will be , which is −∞.
- The limit Does Not Exist.
In order to evaluate the limit as x approaches 6, we find the limit as it approaches 6+ (from the
right) and the limit as it approaches 6− (from the left). If the two limits approach the same
value, or both approach positive infinity or both approach negative infinity, then the limit is that
value, or the appropriately signed infinity. If the two limits do not agree, the limit “Does Not
Exist.” Here, if we look at the solutions to problems 9 and 10, we find that as x approaches 6+,
the limit is +∞, but as x approaches 6−, the limit is −∞. Because the two limits are not the
same, the limit Does Not Exist.
- 1
Here we have to think about what happens when we plug in a value that is very close to 0, but
a little bit more. The top and bottom expressions will both be positive and the same value, so
we get .
