Limits can be used to determine the end behavior of a function. This is done by computing the limit of the function as x approaches both infinity and negative infinity. Which, written mathematically is:
Unlike other limits when computing the limits at infinity, we only see what value the function is approaching from one direction and not both. The reason for this is that we cannot get a value larger than infinity or smaller than negative infinity.
Often times, when computing the limit at infinity, we’ll get the following situations: ∞/∞, k/∞, ∞/k, or k where k is any real number.
If computing the limit you get the value of k/∞, then the value of the limit is considered 0 because the denominator is so much larger then the numerator can ever be.
If computing the limit you get the value of ∞/k, then the value of the limit is considered ∞ because the numerator is so much larger then the denominator.
When computing the limit at infinity and you get ∞/∞ it is considered indeterminate. Remember, ∞ ≠ ∞ because ∞ is not a number but rather a mathematical concept indicating a very large number. So, the question is what value approaches infinity first, the numerator or the denominator, or do they approach at the same rate. L’Hospital’s Rule can be used help us determine what the end behavior of the function is.
When computing the infinite limit and you get some constant value, k, you have found that the function has a horizontal asymptote, which is k.