A limit is the value that a function approaches as its inputs come closer and closer to a certain number. The concept of a limit lies at the heart of all calculus. Without limits, there would be no way to discuss the behavior of functions near their extreme values.

In **calculus classes**, we often use limits to illustrate **important concepts** in both single-variable and multi-variable functions. For example, we might want to illustrate that the rate of change of a function is not always positive by plotting a function and pointing out where it goes from increasing to decreasing or vice versa. We can also use limits to show that a function cannot always be extended beyond a point. For example, we might want to show that there's no way to fit "0"s on an algebraic curve by looking at the graph of $f(x) = x^3$ and noting that if we keep adding **more "0"s** to the curve, the curve will never go back down toward the axis. Limits also help us understand what happens to different parts of a function as its input values get very close together or exactly equal.

Limits play an important role in many problems in calculus. For example, when solving for the maximum or minimum value of a function, we need to know whether the function is increasing or decreasing near the peak or minimum value.

For example, the limit of x as x goes to 3 of y is 5, because no matter how close x gets to 3, it can never be exactly 3. The limit of x cubed as x goes to 0 is 1, because if you try to calculate (0 + 1)3 you'll get 1 7 instead.

In physics, we often want to know what happens to **a physical system** when some parameter reaches **a certain value**. For example, if I throw a ball up in the air, I want to know how high it will go before it comes down. If you ask me how high it went, I could say "It went pretty high", but there's no way for me to know for sure unless I catch it first. However, if we look at the math behind throwing a ball up in the air, we see that the only thing that affects how high it goes is the initial speed of the ball when it's thrown. Once that speed is known, we can always work out how high it will go by using **simple equations** called "laws" of motion.

The limits used in physics are usually very small values.

A limit is the value that a function (or sequence) approaches when the input (or index) approaches some value in mathematics. Limits are used to define continuity, derivatives, and integrals in calculus and mathematical analysis. They also appear in **many other areas** of mathematics.

In computer science, a limit is often used as **a hard parameter** for a function. For example, if we want a function f to be defined for all real numbers, then we can write:

F(x) = x for all x <= 0

This means that the function will output zero for any input less than or equal to zero. We need this because later on we'll want to calculate **y = f(3**). If we didn't specify that the function should only work for positive inputs, then y would get calculated as f(3) for negative values too, which doesn't make much sense. In this case, we need to say that the function should not be called for negative inputs.

Limits are also important in statistics. For example, consider the case where we have **data points** x, f(x) where f is some function and x is some value between 0 and 1. Then, it makes **no sense** to ask what f of 0.5 is, since there are no such things as half-values.

A mathematical notion based on the concept of proximity that is largely used to give values to **particular functions** at locations where no values are defined in order to be consistent with adjacent values. Analysis: Function Continuity...

One of the most crucial concepts to grasp in order to do calculus is the notion of limits and continuity. A limit is defined as a number that a function achieves when its independent variable reaches a specific value. Continuity refers to the property of a function being equal to its limit as its argument approaches a particular value. The idea behind these concepts is that if we can find out when a function becomes equal to its limit, then we can use this information to determine what happens to the function near its limit values.

Limits and continuity are important because they tell us how functions behave when their arguments get close to certain numbers. For example, if we know that a function is continuous at some number, then we can be sure that it will always stay within some distance of **that number**. If we knew nothing about the function except that it was continuous, then we could not make such a claim; instead, we would have to examine each case separately. Limits also help us understand why some functions are easier to work with than others. For example, it might be difficult to calculate the limit of the sequence 1/n, but since e is a constant number, finding its limit is easy. Functions that involve constants or variables that approach certain values can often be calculated explicitly; other functions cannot be solved this way but their limits provide sufficient information for us to estimate what the function is like near **these values**.