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In today’s article on limits in higher mathematics, we skip over the definition of limits and some common limit calculations. I think you’re familiar with some of the more common functions and the limits of sequences.


Most of the simpler functions or sequences, we can see their limits very intuitively. Such asAs n approaches infinity,The limit of omega is zero, and as n approaches infinity,The limit of phi is infinite, and so on. However, for some relatively complex functions, it may be difficult for us to intuitively see the limit, so we need toA more convenient method to calculate the limitToday’s article introduces just such a methodPinch and substitution.


Clamp force method


The pinch method is actually very common in mathematics, and often appears in high school competitions. The principle of the pinch method is very simple. For a certain function f(x), we know its expression, but it is difficult to determine its range. We can first find two other functions g(x) and h(x) whose scope is relatively easy to determine, and then prove:. The range of f(x) is squeezed by the range of h(x) and g(x).


To put it bluntly, we solve inconvenient functions directly, and solve them indirectly by replacing them with other functions that are easy to calculate, similar to “saving the country by the curve”.


Now that we understand the concept of squeezing, let’s look at its application to series limits. There is a sequence of numbersWe need to determine its limit, and we found two more sequencesand. If they meet the following two conditions:


  1. whenThere are.


So, the sequenceThe limit of phi exists, and. Intuitively, this should be pretty straightforward, but let’s just try to prove it mathematically, and review the definition of limits.


The proof process is as follows:


By the definition of limit, for a sequenceFor anyThere are, so that for any:, there are. So it’s called a sequenceThe limit of theta is a.


Due to the sequenceThe limit of phi is a, so it existsmakeWhen,. In the same way, there ismakeWhen,. So forObviously there should be:and.


If we expand the absolute value, we can get:


We are in, can be obtained:


By definition of the limit, obviously you get a sequenceThe limit of alpha is also a.


Let’s use this method to look at an example from the book, and we all know that as x approaches 0,andThey all go to 0, butWhat’s the limit of theta? If you guess, the ratio of two limits as they go to 0 should be 1, but that’s just an intuitive guess, and if you want to prove it strictly, you have to do it mathematically.


This proof uses what we just said about squeezing, and it’s very clever, so let’s look at the picture below.

Let’s say the AngleThe radian system is used here. Let’s set the length of the center OB equal to 1, so... So we’re going to use this oneThe area relation of geometric figures, it is clear that:


< the area of sector AOB <In the area.


The area of delta is equal to.The area of delta is equal to. Both of these are easy to figure out, just by applying the formula for the area of a triangle. The area of the fan looks a little bit more complicated, but it’s actually very simple, and in geometry,A fan can be thought of as a special triangle. If we view the arc length as the base, and we can view the radius as the height, then the area of the sector is equal to. So the area of the fan AOB is equal to.


If we list them, we can get:


That is:


Among themSo we can divide both sides of the inequalityTo get:


Because as x approaches 0Both are greater than 0, so we can interchange the numerator and denominator of the inequality, and get:


I’m done here, because it’s easy to see from the graph of cosine, as x approaches 0, cosine of x approaches 1. But for the sake of rigor, let’s pretend we don’t know this, and go ahead and prove it mathematically:


So let’s figure out as x approaches 0,As x approaches 0, so. We have toThe transformation is going to be introduced hereThe formula for the sum differential product of trigonometric functions:


Due to the, and the differential product can be obtained:


And we’ve already shown by the area notation that as x approaches 0, so. As x approaches 0, obviouslyIt also goes to 0, so we can prove thatThe limit of theta is 1.


Change element method


Let’s move on to substitution, which is the scientific nameLimit algorithm for composite functions. The definition is as follows: Suppose we have, we make. ifAnd as x approaches PIThere are, then:


It’s also easy to prove it by using the definition of a limit, but I won’t prove it here, so if you’re interested, you can try to prove it.


Now that we know the limit rules for conforming functions, let’s do one more example to reinforce that.


Similar to the above example, let’s solve this time:.

As in the above problem, we first use the sum differential product to transform the limiting molecule, which can be obtained:



And if you calculate this by the definition of the limit itself, it’s kind of complicated, and it’s hard to get an intuitive answer. And that’s where the substitution comes in. Let’s say, then this limit can be converted to the limit of the composition function.. Because as x approaches 0, u also approaches 0, and as u approaches 0,Approaches 1, so the final limit is 1.


By means of the squeezing method and the limit substitution formula of the composite function, we can easily solve some seemingly tricky limits. And this is something that we use a lot when we’re trying to figure out limits. Although the above formula looks a bit troublesome, but the method itself is not difficult, as long as you put your heart into it, you can certainly understand.


Original is not easy, I hope my article can bring you harvest. Scan my official account to get more articles. Your support is my biggest motivation.

The resources

Advanced Mathematics, Tongji University, sixth edition

Programmer math