The derivative is one of the most important concepts in higher mathematics. The derivative is the rate at which a function is changing.
For example, if f(x) represents the distance someone is walking as a function of time, then the derivative would be the rate at which their position is changing (their movement).
This would be expressed as: \frac{df}{dt}=m, where m represents their average movement over a certain amount of time (dt).
More formally, the derivative of a function f(x) is: \frac{df}{dx}=f’(x), where ‘denotes the little arrow traditionally used to designate an implicit differentiation. x represents any given input value.’
This article will discuss how to find the limit of a function by representing it as the derivative of some other function. This will be discussed in detail below.
Definition of a limit
When a function approaches a constant value as the independent variable approaches a specific value, the constant value is called the limit.
In formal terms, the limit of f(x) as x approaches a is L, written as:
Limit = L If f(x) ≈ L when x ≈ A, then we write this as: lim 𝑟→A 𝑟 ≈ L.
Note that in this case, A does not have to be a specific value of x. It can be any arbitrary value within the domain of x. The only requirement is that it must be approached in such a way that f(x) and L are equal when x equals A.
The limit of a function at a certain number is simply some number or constant valued quantity. More formally, the limit of f(x) as x approaches A is L if and only if: f(a)≤L and for every ε>0 there exists some δ>0 such that |f(x)|≤εifx≤δ
Selfie
Another note on limits is that they are not necessarily constants. In fact, most limits are actually functions!
- Example 1: Consider lim 𝑟→1 𝑟 − 1
- Example 2: Consider lim 𝑟→1 (1+𝑟)/((1+𝑟)*√5−1)
Intervals Represent Some Number A to Some Number B.
(Like I said before, intervals can also contain other numbers besides just two endpoints.) Interval arithmetic (addition/subtraction/multiplication/division/equality testing) works just like real arithmetic!
- Example 3: Consider [−4,-2] + [-2,-4] = [−4,-6], so [-4,-6]=[−4,-2]+[−2,-4] and thus [-6]=[−4]+(-2). Since [-6] isn’t defined yet(-6 isn’t in R), we take the least common multiple of [-4] and (-2), which is 6.
- Example 4: Consider [3k+1,[5k+7][9k+11]. In order to add these three numbers together you first have to find their least common multiple which would be 30k+10k+9k+. Then add them together! The answer would therefore be ([30k+(10k+(9k+)[[[[[[30kkkkkkkkkkkkkkkmmmmmm]]]]]]]]]]] + ([30kmmmm]+[30kmmm]) == [[{30kmmmm}m{{{}}}}{{{}}}{{{}}}{{difference{{\rm between{\rm }}these{\rm }}\rm }}([60kmmm],[90kmmm])]. Note how we didn’t have to worry about adding all those zeroes! We just treated them like 0s since their LCM was 0.
Examples of limits
Some common limits are zero, infinity, and negative infinity. Zero represents a point where the function values are constant, infinity represents a point where the function values increase without bound, and negative infinity represents a point where the function values decrease without bound.
Any of these can be represented in any direction: up, down, to the left, or to the right. When representing these in mathematics, you use either plus or minus signs to indicate which way the value changes at these points.
Limits can also be expressed as fractions or decimals instead of integers. For example, the limit of 3/2*x when x increases indefinitely can be expressed as 1.5. There are many ways to express a limit! Some more common ones are described below.
Limits can also be expressed as epsilons and deltas (lowercase Greek letters). These represent small changes in the input variable that result in small changes in the output variable.
Limits and infinity
When the derivative is zero, this means that the function is constant. A constant function represents any value, so the graph of a constant function is a single line with any value.
As we saw in the first section, when you divide by zero, you get an undefined answer. This means that there is no graph for this situation.
So, how does one define the graph of a constant function? By declaring that its derivative is also a constant function! This makes sense- if the graph does not change at all as you move along the axis, then it must be constant.
Limits are very important to understand when dealing with calculus. A limit refers to either infinity or some other large number. When calculating calculus problems involving limits, you must be careful not to overshoot and enter a value that is too high or too low for your answer.
Limits and algebra
Another interesting connection between limits and algebra is that you can use algebra to determine whether or not two functions are equal at a certain point.
If you know the values of the variables in a function, you can determine if two functions are equivalent by solving an equation. For example, if you know the value of x in some function f(x) and the value of x in some other function g(x), then you can solve an equation containing f(x) and g(x) to find out if they are equal at the same point.
This is because solving an equation equals 0 means that something is equal to zero, which means that something else must be equal to it. If something else is not equal to zero, then the things are not equal.
You can also use limits to determine whether or not two functions are equivalent at a certain point.
Understanding the limit equation
The limit of a function as it approaches a particular value is the value the function approaches as accuracy increases.
For example, if you were to draw a line one centimeter long, then draw another line that was one millimeter long, then another that was one tenth of a millimeter long, and so on, you would eventually reach a point where your line was infinitesimally short.
This infinitesimal length is called a vanishing point, and it represents the number 0. At this point, your line becomes nonexistent and there is no longer any length to it.
This idea applies to functions as well. As you increase the accuracy of your input values (that is, use numbers that are closer to the true value), then there comes a point where your function becomes zero at that given input.
Solving for the limit value
Sometimes, you will be asked to solve for the limit value instead of just determining if the limit exists or not. To solve for the limit value, you have to find the derivative of the variable at some number A and then see if it equals L or not.
For example, let’s say we are trying to solve lim x->0 x^2 – 2x + 1 = 0.
To solve for the limit, we have to first find the derivative of x^2 – 2x + 1 at 0 (which is (-1) ^0 / (2 * 1) = -1). Then, we have to see if it equals 0 or not.
This problem is a little more complex because we are solving for a negative number. Since we are trying to solve for -1, we can conclude that x^2 – 2x + 1
The closer the number is to zero, the closer the limit is to infinity
So, for example, if we were to find the limit of 1/x as x approaches 0, then 1/x would get closer and closer to zero as x gets closer to zero, but it would not actually reach zero.
This is because as x gets closer to zero, 1/x gets closer and closer to infinity. The ratio between the two values becomes larger and larger, so the ratio between zero and any other number becomes larger and larger.
In other words, the value of 1/x does not stabilize at some finite value as x approaches 0. The limit of 1/x as x approaches 0 is infinite. This is called lim(1/x)=infinity. Infinity is called a special value of a function.
Limits are useful in many applications
The most common application of limits is the differentiation algorithm. This is a method for computing the derivative of a function, f(x), which is a ratio of two variables, x and f(x).
The derivative is defined as the limit of the difference between f(x) and x as x approaches 0. Since we are considering how close x gets to 0, we call this variable t. The derivative is written as:
f”(x) = lim t→0 (f(x+t)−f(x))
This definition can be understood by looking at an example. Let’s take the function f(x) = 2x + 4. Then, the derivative f”(x) = 2. By substituting t = 0 into the definition of the derivative, we get that f”(0) = 2·4 − 2·0 = 4 − 0 = 4. This makes sense because when x equals zero, then (2x + 4)−>2·0=4.
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