When discussing curves, you want to get as much information as possible about a function and its graph. The so-called limit value provides information about how the y-values behave when the x-values go in a certain direction. The limits are thus an important topic in the field of functions in mathematics. In this article you will find out what you should definitely know about the limit value. Have fun with your studying!
What is a limit?
In mathematics, the limit value of a function at a specific point is the value that the function approaches in the vicinity of the point under consideration. In mathematics, limit values are always used when you want to examine the behavior of a function in the vicinity of an x-value that you cannot use in the function yourself.
However, such a limit does not exist in all cases. If the limit exists, the function converges, otherwise it diverges. The limit concept was formalized in the 19th century and is one of the most important concepts in analysis.
The limit values can be specified using the Limes. The limit describes what happens when you use values for a variable that get closer and closer to a certain value. Under the «lim» is the variable and against which number it goes, i.e. which value the variable is getting closer to.
After the «lim» comes the function into which the values for x are inserted. This can then look like this, for example:
This notation means that values are used for x in the function 1x, which are getting closer and closer to infinity. One then speaks “Limes towards infinity”. This procedure also works with all other values.
determine limits
To determine the limit value, one can distinguish different cases, which I will now go into in more detail.
limits at infinity
To illustrate this topic, let’s consider the graph of a normal parabola. The graph of the function f(x)= x2 is drawn in the coordinate system.
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Limit values at infinity describe what happens to the function, i.e. to which value the function approaches as x approaches infinity. In doing so, x can run towards + and – infinity, i.e. it can always become smaller or larger. In mathematical notation, this looks like this:
and
The limit value then looks graphically as shown in the figure. If you want the limit for +∞ or -∞, see what the function «does in the direction». Here it goes in both directions towards infinity.
To examine how the y-values behave as the x-values get larger and larger, one can set up a table of values:
x
1
10
100
1,000
….
f(x)
1
100
10,000
1,000,000
….
It can be seen that the function values become infinitely large. Mathematically formulated this means:
How the y-values behave when the x-values get smaller and smaller can be determined very easily in the same way by letting the limit run towards minus infinity.
Limits at a finite point
Limits in the finite are values that the function takes on as it approaches a given value. This is often used on definition gaps to check what is happening near that gap. You can approach the value from the left or right, i.e. approach the definition gap either from the negative side or from the positive side. This can result in different limit values.
It is all noted as follows:
and
Instead of x → ∞, we are dealing here with x → x0. where x0 is a real number.
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Limits of functions consisting only of polynomials
How do you calculate the limit of a function if the function consists only of polynomials? If there are only polynomials in the function, you first determine the x with the highest exponent. If you let x go towards +∞ or -∞, other parts of the function can never grow as large as this term. Therefore it is sufficient to consider only the term containing the x with the highest exponent.
So if you have the function, it’s enough, just the
consider.
Limit values for function jumps and definition gaps
Function jumps and definition gaps can be approached from the left or right, the limit values are different in each case. A function jump occurs when there is a case distinction in the function rule. This is indicated by a quantity notation, for example like this:
On the figure you can see the corresponding function value A at point a. If you approach this function jump from the left, the limit value is B.
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If you want to calculate the limit value of the function at the function jump from the left,
so you write:
However, if you approach from the right, you use the following notation:
The definition gaps can also be approached from the left and right.
A more precise method for determining these limits would work via a corresponding sequence that converges to zero, e.g. the sequence . You would then use this together with the a in the function and let it run to zero, for example by letting n run to infinity.
Limits for specific functions
For the sake of completeness, here are the limit values for certain functions, namely for the power functions and the exponential functions.
The limit of a power function is given by:
(Source: studimup.de)
For exponential functions, the limit is given by:
(Source: studimup.de)
Limit values - everything important at a glance
Well, have you reached the end of the article? At the end of the topic you will get an overview of the most important aspects of the limit value so that you are well prepared for the next test.
- In mathematics, the limit value of a function at a specific point is the value that the function approaches in the vicinity of the point under consideration. It is an important key figure in the context of a curve discussion.
- The limit values can be specified using the Limes. It describes what happens when you substitute values for a variable that get closer and closer to a certain value.
- Limit values are usually determined mathematically with the help of value tables.
- The limit at infinity reveals how the y-values behave as the x-values become larger and smaller.
- The limit at a finite point tells how the y-values behave as the x-values approach the point x0.
- The limit of a power function is .
- For the limit of an exponential function applies.