Limits of Algebraic Functions: Concepts, Methods, Problems and Their Discussion

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Limits of Algebraic Functions – Does Sinaumed’s realize that in living his daily life, it is also closely related to mathematical concepts? Not only on the basic calculation concept, but even on the limit function concept. When walking through the toll road, did Sinaumed’s ever casually look at the straight highway in the distance? Then see the vehicles that cross us moving farther and farther and getting smaller in size. Well, that indicates that we have a limit. Not only in sight, but there is also a threshold for hearing, a limit on the ability to carry a burden, a limit on the ability to buy an item, and others.

If in mathematics, this limit is called the term “limit”. The limit function can be related to several other branches of mathematics, including algebra and trigonometry. Now, this time we will discuss the limits of algebraic functions. What is the limit of an algebraic function? What are the properties of the limit of this algebraic function? What is the method of solving the limits of this algebraic function? Come on, look at the following reviews so that Sinaumed’s understands these things!

What is the Limit of an Algebraic Function?

Basically, limit is a value that uses a function approach when trying to approach a certain value. In short, this limit is considered as a value towards a limit. It is referred to as the “limit” because it is indeed ‘close’ but cannot be reached. Then, why should this limit be approached? Because a function is usually not defined at certain points. Even though a function is often not defined by certain points, it is still possible to find out what value the function can approach, especially when a certain point is getting closer to the “limit”.

Limit formula

That is, if x approaches a but x is not equal to a, then f(x) will approach L. This approach of x to a can be seen from two sides, namely the left side and the right side. So, in other words, x can also approach from the left and right so that later it will produce a left limit and a right limit.

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Therefore, the statement is obtained that:

0 <|xp|<δ⇔|f(x) – L|ε

That is, a function can be said to have a limit if the left limit and right limit also have the same magnitude. If the left limit and right limit are not the same, then the limit value will also not exist.

Properties of Algebraic Limit Functions

If n is a positive integer, k is a constant, f and g are functions that have a limit in c, then the properties will be:

Methods in Solving Limits of Algebraic Functions

There are several simpler methods for determining limits, namely by substitution, factoring, and rationalizing the denominator. How are these methods applied, let’s look at the following review!

1. Determining Limits with Substitution

If the value of a function for x approaches a, where a is a real number, then it can be determined by means of substitution. In this substitution method, the value of x will be replaced with a. However, if the result becomes (∞-∞) or 0/0 ∞/∞. So this method cannot be applied directly. It’s better if the function that takes the limit needs to be simplified again. Consider the following example.

the result of limit   is 1.

By using the substitution method, the limit value is obtained as follows:

Thus, the product of limit   is 1.

2. Determining Limits by Factoring

In this way, let’s say we have a problem of the form lim →af(x)/g(x)  . Now, as explained earlier, if x = a then it can be substituted for the function that takes the limit, so that it will produce

Therefore, the function must be simplified again by factoring f(x) and g(x) so that they both have the same factor. Furthermore, the same factor can be removed so that it will obtain an even simpler form, as follows:

3. Determine Limits by Rationalizing the Denominator

In this method, if in a function to be determined the limit value turns out to be difficult to simplify because it contains an irrational denominator, then it can be solved by rationalizing the denominator first. We learned how to rationalize the denominator of a fraction when we were in elementary school, do you still remember Sinaumed’s ? So, pay attention to the discussion below:

How to Determine the Limit Value of an Algebraic Function?

1. How to determine the limit of an algebraic function if the variable is close to a certain value

If you encounter an algebraic limit problem where the variable approaches a certain value, it can be solved using several methods, namely:

  • Substitution Method

Look at the following example questions!

Determine the value of lim 2x 2  + 5 x →3

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Completion:

So when asked what is the limit value for the function above?

We replace the value x = 3 for the variable x at 2x 2 , now this is called substitution. So the solution to the limit above by substitution is:
lim 2x 2  + 5 = 2.(3) 2  + 5 = 23 x →3

  • Factoring Method

This method will be used if the functions can be factored so that they do not produce undefined values. Check out the following examples!

Now, Sinaumed’s must have known that any number divisible by 0 would be undefined. That means if we determine the value in the example problem earlier, we have to find a new function so that division by 0 does not occur.

Using the substitution method will produce an undefined form (0/0):

Then it must be solved by the factoring method:

  • The Method of Rationalizing the Denominator

This third method can be used if the denominator is in the form of a root which really needs to be rationalized, so that the division of the number 0 by 0 does not occur. Consider the following example problem!

Example:

  • The Method of Rationalizing the Numerator

In this method, it is almost the same as the previous method, which can be used if the denominator is in the form of a root which really needs to be rationalized, so that the division of the number 0 by 0 does not occur. Consider the following example problem!

  1. Determining the Limit of an Algebraic Function If the Variable Approaches Infinity

The limit form of an algebraic function can also occur if the variable approaches infinity, for example like:

lim x→∞ f(x)/g(x) lim x→∞ [f(x)+g(X)

So, if there is such a problem, it can be solved using several methods, namely in the form of dividing by the highest rank and multiplying by the opposing factor. Here’s the review! lim x→∞ f(x)/g(x) lim x→∞ [f(x)+g(X)

  • The Method of Dividing by the Highest Rank

In this method, it is usually used to find the value of lim x→∞ f(x)/g(x) . The trick is to divide f(x) and g(x) by the highest power of n contained in f(x ) or g (x). To understand better, consider the following examples!

Problems example :

  • The Method of Multiplying by the Opposite Factor

In this method, it is used to solve lim x→∞ {F(x)+G (X)

Look at the following examples of questions and solutions!

So, that’s a review of what is the limit of an algebraic function and the methods that can be used to solve it. Has Sinaumed’s applied some of these methods to this algebraic limit problem?

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