Limits: Behavior Of Functions As X Approaches Infinity

limits as x approaches infinity

horizontal asymptotes

When we say limits as x approaches infinity, we are essentially asking the question: What value does the function approach as x gets larger and larger? The answer to this question depends on the function in question.

There are three types of behavior that a function can exhibit as x approaches infinity:

1. The function approaches a finite number:
This happens when the function settles down to a constant value as x gets larger and larger. For example, consider the function f(x) = 1/x. As x gets larger and larger, f(x) gets closer and closer to 0, but it never actually reaches 0. In this case, we say that the limit of f(x) as x approaches infinity is 0.

2. The function approaches infinity:
This happens when the function grows without bound as x gets larger and larger. For example, consider the function g(x) = x^2. As x gets larger and larger, g(x) grows without bound, so we say that the limit of g(x) as x approaches infinity is infinity.

3. The function oscillates:
This happens when the function does not approach a finite number or infinity as x gets larger and larger, but instead oscillates between different values. For example, consider the function h(x) = sin(x). As x gets larger and larger, h(x) oscillates between -1 and 1, so we say that the limit of h(x) as x approaches infinity does not exist.

In general, to determine the limit of a function as x approaches infinity, we can look at the behavior of the function as x gets larger and larger, and use algebra or calculus techniques to simplify the function and identify its limiting behavior.

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