Limits give all mathematics students a lot of trouble. To solve a limit, sometimes you have to use a lot of tricks and choose from a variety of solution methods exactly the one that is suitable for a particular example.

In this article we will not help you understand the limits of your capabilities or comprehend the limits of control, but we will try to answer the question: how to understand limits in higher mathematics? Understanding comes with experience, so at the same time we will give several detailed examples of solving limits with explanations.

The concept of limit in mathematics

The first question is: what is this limit and the limit of what? We can talk about the limits of numerical sequences and functions. We are interested in the concept of the limit of a function, since this is what students most often encounter. But first, the most general definition of a limit:

Let's say there is some variable value. If this value in the process of change unlimitedly approaches a certain number a , That a – the limit of this value.

For a function defined in a certain interval f(x)=y such a number is called a limit A , which the function tends to when X , tending to a certain point A . Dot A belongs to the interval on which the function is defined.

It sounds cumbersome, but it is written very simply:

Lim- from English limit- limit.

There is also a geometric explanation for determining the limit, but here we will not delve into the theory, since we are more interested in the practical rather than the theoretical side of the issue. When we say that X tends to some value, this means that the variable does not take on the value of a number, but approaches it infinitely close.

Let's give a specific example. The task is to find the limit.

To solve this example, we substitute the value x=3 into a function. We get:

By the way, if you are interested, read a separate article on this topic.

In the examples X can tend to any value. It can be any number or infinity. Here's an example when X tends to infinity:

Intuitively, the larger the number in the denominator, the smaller the value the function will take. So, with unlimited growth X meaning 1/x will decrease and approach zero.

As you can see, to solve the limit, you just need to substitute the value to strive for into the function X . However, this is the simplest case. Often finding the limit is not so obvious. Within the limits there are uncertainties of the type 0/0 or infinity/infinity . What to do in such cases? Resort to tricks!

Uncertainties within

Uncertainty of the form infinity/infinity

Let there be a limit:

If we try to substitute infinity into the function, we will get infinity in both the numerator and the denominator. In general, it is worth saying that there is a certain element of art in resolving such uncertainties: you need to notice how you can transform the function in such a way that the uncertainty goes away. In our case, we divide the numerator and denominator by X in the senior degree. What will happen?

From the example already discussed above, we know that terms containing x in the denominator will tend to zero. Then the solution to the limit is:

To resolve type uncertainties infinity/infinity divide the numerator and denominator by X to the highest degree.

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Another type of uncertainty: 0/0

As always, substituting values into the function x=-1 gives 0 in the numerator and denominator. Look a little more closely and you will notice that we have a quadratic equation in the numerator. Let's find the roots and write:

Let's reduce and get:

So, if you are faced with type uncertainty 0/0 – factor the numerator and denominator.

To make it easier for you to solve examples, we present a table with the limits of some functions:

L'Hopital's rule within

Another powerful way to eliminate both types of uncertainty. What is the essence of the method?

If there is uncertainty in the limit, take the derivative of the numerator and denominator until the uncertainty disappears.

L'Hopital's rule looks like this:

Important point : the limit in which the derivatives of the numerator and denominator stand instead of the numerator and denominator must exist.

And now - a real example:

There is typical uncertainty 0/0 . Let's take the derivatives of the numerator and denominator:

Voila, uncertainty is resolved quickly and elegantly.

We hope that you will be able to usefully apply this information in practice and find the answer to the question “how to solve limits in higher mathematics.” If you need to calculate the limit of a sequence or the limit of a function at a point, and there is absolutely no time for this work, contact a professional student service for a quick and detailed solution.

Know that everyone has a different “hurt”,
Know that everyone has a different “scary”.
Don't judge from your bell tower
It is unknown how many floors there are.

Don't look at the boundaries
Don't invent limits with your brain.
What would you never dream of in a nightmare?
For some it’s a common thing.

Know that everyone has a different “should”,
Know that everyone’s “difficulty” is different.
However, the idea of hell
It is impossible to generalize and compare.

Know that the truth is different,
And not the one that is served on a platter.
Take a closer look at those whose fates are frightening,
These are the strongest people.

Don't say I don't remember you...
I remember everything, many times a day
I repeat the phone number
But I will never call you.
It felt like my heart was about to break
And at the limit of lonely days

They went beyond the horizon in my soul.
There was love, there was love, there was!
And there is nothing to add to this phrase.
Our magic castle burned to the ground
And he didn’t leave ashes for us to remember.
I remember everything, and I remember the blooming garden,
And through the foliage - rays from all sides,
As if from a white, white bell tower
In your soul - you hear - a quiet ringing is pouring.
Love is gone and will never return,
And so as not to miss her forever,
Your eyes are like two sad suns
They went beyond the horizon in my soul.

The pursuit of happiness is again unsuccessful...
And it’s a rainy evening, it’s gloomy outside...
And as a child... I spread jam on a bun
And definitely happy, stupidly happy...

Glamour, etiquette, diamonds, jacuzzi...
Now, in addition to happiness, in the fate of “All inclusive”,
And as a child I ate sunflower seeds,
And happiness seemed to have no limits...

We have become very much like clowns...
Everyone is wearing makeup that laughs on the outside...
And in childhood... only the sun broke through from the sky
And the happy heart smiled so much...

We select people like buckwheat in Cinderella...
Contact everyone you need... Those who are unprofitable go to the stove...
And in childhood, clear skies believed in us...
Where is the joy from the smell of fresh bread?

And now friendship can be bought too...
We made it... We live in a world of fur and leather...
And as a child, they saved a mongrel from a rainstorm...
And by giving happiness, they received it.

We have lost sincerity and sensitivity over the years...
We came up with boundaries and frames ourselves...
Do you have a bun and a jar of cherry jam?
So be happy as hell!

I look at you and realize that I still love you. This love is a chronic disease of recent years. It brings such unbearable pain that I throw myself at complete strangers, trying to be deceived by them, with them suddenly in these embraces I will find the very painkiller that, according to the owners of wrinkled hearts, does not exist at all. I understand that I’m being deceived, but I still continue to hug - I can’t kill myself otherwise, it hurts, it’s tormenting, it doesn’t let me sleep at night, so I’m sitting on the windowsill and, in another minute, I’ll scream heart-rendingly from the torture of illusions. Should I contact you for help? Useless. You know about my love, but you don’t need it, “your mouth is full of unexpressed feelings.” We are in the same web of irresponsibility, but we cannot help each other. You wrap your hands around the thin white threads and rods and look somewhere beyond the boundaries of reality, hoping for God knows whose help. And there is only one difference between us: my love for you almost knocked me off my feet, and your love for someone fuels and revives you with expectation, albeit deceptive. I don’t want to look at you anymore, I drive the possibility of you out of my heart, but this makes it even more painful. So he goes through the process of suffering in a whisper, also hoping for God knows whose help. Time?..

Your life is a complete lie, porn, everyday life, Internet addiction and cell phone slavery. Well, am I not right? Tell me, have you ever done anything truly out of the ordinary? Never. And you can't. Do you know why? Because all of this is outside your comfort zone. You are packed into it. Like in a polypropylene bag. You are pieces of meat, squeezed by the confines of everyday life and work. Or am I wrong? Maybe I'm wrong? Correct me.
For example, can you give your mobile phone to the first person you meet? A? Tricky question. Can you format the screw on your computer right now? Dumb? Did you shit yourself? Do you know why you don't do this? Because it is tantamount to suicide. You don't exist without it.

He has been twenty-eight long years, and twenty-eight winters besides,

He kept one secret and was an exemplary husband in the family.
Everything seemed to be as usual: my wife was preparing dinner...
But suddenly something bad happened: he remembered the secret.
To the noise and sour smell of cabbage soup, the grumbling of my betrothed in the morning,
He remembered everything down to the smallest detail, as if it had happened yesterday...
...She sat by the window, and the soft wonderful moonlight
Painted her beautiful silhouette in pale tones...
Strands flowed over the shoulders, slid like snakes onto the chest...
And he thought rashly: “I’ll marry her someday!”
He remembered everything down to the smallest detail: the curves of the lines, the softness of the lips...
And the heat of her simple speeches, and a huge oak tree outside the window.
A tangle of hands... A merging of bodies... A cascade of brown hair...
And how he wanted her to the point of frenzy, to tears!
A flow of tremulous confessions, how he whispered them in my ear!
A funny curl above the ear that trembled with breathing...
She looked at him with eyes wet as night.
The words were as intoxicating as wine: “I love you... Give me a daughter...”
In the morning he lost peace: he was fussing, then bored...
Then, covering his face with his hand, he sat on a chair and was silent.
The wife grumbled, as always. She scolded the runaway soup...
And he noted that years did not suit her as she aged.
Just as white color and strands of dyed hair don’t suit her.
And for twenty-eight years everything somehow wasn’t serious...
Suddenly he jumped up, grabbed his coat, and forgot about his hat and socks.
All twenty-eight years are not the same... All twenty-eight winters are melancholy.
Found that house. The house has an oak tree. He ran up the stairs like an arrow...
I wish I could stop the trembling from my cold lips, and the disgusting cowardice - away!
She's probably drinking tea and wrapping herself in a shawl...
And a quiet sadness flows from her beautiful eyes...
Or maybe she started knitting? Or maybe weave lace?
There is so much to tell her! And the main thing is to say - I’m sorry...
She opened the door... There was a question in her eyes. She was twenty years old again...
A cascade of brown hair... A silhouette familiar to the heart...
There is a slight curl above the ear... Like many years ago - exactly the same...
" You are not mistaken?" - No, I couldn’t... Are you Anya? “Vera.” Her daughter…"
“And Anya?” - “Mom is no more... Who are you?” He turned back:
“I went to her for twenty-eight years...” - She was waiting for you... Twenty-five...
How dizzy... How my heart sank in my chest!
And he remembered her words with a prayer: “Don’t leave!”
He hunched over. He trudged away. Intertwining of hands... Merging of bodies...
I love you... Give me a daughter... But he really wanted a daughter.
How strange. Anya is no more... I cried... I threw it into the silence: “I will love you for many, many years... You alone...”

P.S. TAKE CARE OF LOVE - it is the foundation of your happiness...