2.2: Limit of a Function and Limit Laws
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.film film come
In mathematics , the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f x to every input x. We say the function has a limit L at an input p : this means f x gets closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to any input sufficiently close to p , the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, we say the limit does not exist. The notion of a limit has many applications in modern calculus.
We want to give the answer "2" but can't, so instead mathematicians say exactly what is going on by using the special word "limit". So it is a special way of saying, "ignoring what happens when we get there, but as we get closer and closer the answer gets closer and closer to 2". Limits can be used even when we know the value when we get there! Nobody said they are only for difficult functions. So instead of trying to work it out for infinity because we can't get a sensible answer , let's try larger and larger values of x:.
As x tends to plus infinity f x gets closer and closer to 0. The limit of a function. The definition of the limit of a function. A limit o n the left a left-hand limit and a limit o n the right a right-hand limit. Continuous function. Limits at infinity or limits of functions as x approaches positive or negative infinity.
Limit of a function
This process is called taking a limit and we have some notation for this. The limit notation for the two problems from the last section is,. In this section we are going to take an intuitive approach to limits and try to get a feel for what they are and what they can tell us about a function. With that goal in mind we are not going to get into how we actually compute limits yet. We will instead rely on what we did in the previous section as well as another approach to guess the value of the limits. Both approaches that we are going to use in this section are designed to help us understand just what limits are. We will look at actually computing limits in a couple of sections.