In calculus, we often encounter scenarios where the input to a function approaches infinity or negative infinity. These situations are crucial to understand, as they help us determine how a function behaves at its extreme ends.

To evaluate limits at infinity, we examine what happens to the function as the input values become arbitrarily large or small. For functions with a dominant term or highest power of x, the limit can be determined by evaluating the behavior of that term. Let's consider an example:

lim(x -> ∞) (3x^2 - 2x + 1) / (x^2 + 5)

In this case, as x approaches infinity, the terms with lower powers become negligible compared to the highest power term, which is the x^2 term. Therefore, we can simplify the expression by considering only the highest power terms:

lim(x -> ∞) (3x^2) / (x^2)

By canceling out the x^2 terms, we find that the limit of the function as x approaches infinity is 3.

In addition to limits at infinity, we also encounter infinite limits. These occur when the function produces an unbounded result (either positive or negative) as the input approaches a particular value. For instance, consider the function:

lim(x -> 1^-) 1 / (x - 1)

As x approaches 1 from the left (represented by the negative superscript), the denominator becomes infinitely small while the numerator remains constant at 1. Hence, the result approaches negative infinity, indicating an infinite limit.

Remember, these concepts can seem perplexing at first, but with practice and exposure to a variety of examples, you'll develop an intuition for limits at infinity and infinite limits!