Function limits

Aim: If any function isn't defined for some value of the argument tema6_EN_files\tema6_EN_MathML_0.jpg, then it is natural to inquire what is the behavior of this function when argument tema6_EN_files\tema6_EN_MathML_1.jpg assumes values infinitely attend totema6_EN_files\tema6_EN_MathML_2.jpg. For this reason it is necessary to set up and adopt the concept of a limit. The other aim is to form skills for finding the limit of a function.


Knowledge and skills to find limits of functions leads to:

1. More precise and exhaustive constructions of function graphcs.

2. Capability to find independently the derivative of a function.

Presentation structure:

1. Why is the concept of function limit studied.

2. Rules for finding limits. Examples.

3. Limits of some of the basic functions. Illustrations.

4. Several examples with solutions.

5. Examples for individual work.

1  Why is the concept of limit studied

The answer to this question isn't simple that is why we begin with several examples. Let us examine the function tema6_EN_files\tema6_EN_MathML_3.jpg It is evident that this function isn't defined at tema6_EN_files\tema6_EN_MathML_4.jpg. Then let us consider what is the behavior of the function when argument tema6_EN_files\tema6_EN_MathML_5.jpg takes values infinitely close to the point tema6_EN_files\tema6_EN_MathML_6.jpgThe proximity with this point has two aspects: left proximity (marked using the symbol "1-0") and right proximity (marked with the symbol "1+0").

Let there be given a different function tema6_EN_files\tema6_EN_MathML_7.jpg, which is defined for everytema6_EN_files\tema6_EN_MathML_8.jpg. The important thing is to know what the behavior of this function is when tema6_EN_files\tema6_EN_MathML_9.jpgand tema6_EN_files\tema6_EN_MathML_10.jpg. But should we encounter a function tema6_EN_files\tema6_EN_MathML_11.jpg, which is defined for tema6_EN_files\tema6_EN_MathML_12.jpg, then we would have to be aware what values the given function would approach when tema6_EN_files\tema6_EN_MathML_13.jpg (to the right of tema6_EN_files\tema6_EN_MathML_14.jpg) and tema6_EN_files\tema6_EN_MathML_15.jpg(to the left of tema6_EN_files\tema6_EN_MathML_16.jpg).

Everything mentioned up to this point suggests that there is the need to bring in the concept of limit of functionwhen argument tema6_EN_files\tema6_EN_MathML_17.jpg approaches some point tema6_EN_files\tema6_EN_MathML_18.jpg. The symbol used is

tema6_EN_files\tema6_EN_MathML_19.jpg (tema6_EN_files\tema6_EN_MathML_20.jpg is read as limit).

The above written means that the number tema6_EN_files\tema6_EN_MathML_21.jpg is the limit of the function tema6_EN_files\tema6_EN_MathML_22.jpg, when tema6_EN_files\tema6_EN_MathML_23.jpg (to the left of right of tema6_EN_files\tema6_EN_MathML_24.jpg). The value for tema6_EN_files\tema6_EN_MathML_25.jpg can be either tema6_EN_files\tema6_EN_MathML_26.jpg or tema6_EN_files\tema6_EN_MathML_27.jpg. The same is also true for tema6_EN_files\tema6_EN_MathML_28.jpg

Example 1.1. Find the limit of the function tema6_EN_files\tema6_EN_MathML_29.jpg, when tema6_EN_files\tema6_EN_MathML_30.jpg(to the left and right) and the limit of the function tema6_EN_files\tema6_EN_MathML_31.jpg, when tema6_EN_files\tema6_EN_MathML_32.jpg (to the right tema6_EN_files\tema6_EN_MathML_33.jpg and to the left tema6_EN_files\tema6_EN_MathML_34.jpg)



Comment: Do not take symbols tema6_EN_files\tema6_EN_MathML_39.jpg or tema6_EN_files\tema6_EN_MathML_40.jpgas zero division, but as division by an infinitely small positive tema6_EN_files\tema6_EN_MathML_41.jpg or negative tema6_EN_files\tema6_EN_MathML_42.jpg number.

Example 1.2. Find the limits of the functions tema6_EN_files\tema6_EN_MathML_43.jpgand tema6_EN_files\tema6_EN_MathML_44.jpg, when tema6_EN_files\tema6_EN_MathML_45.jpg.


tema6_EN_files\tema6_EN_MathML_46.jpg tema6_EN_files\tema6_EN_MathML_47.jpg tema6_EN_files\tema6_EN_MathML_48.jpg tema6_EN_files\tema6_EN_MathML_49.jpg

Comment: Symbols tema6_EN_files\tema6_EN_MathML_50.jpg or tema6_EN_files\tema6_EN_MathML_51.jpgtake as division by an infinitely big positive tema6_EN_files\tema6_EN_MathML_52.jpg or negative tema6_EN_files\tema6_EN_MathML_53.jpg number.

It isn't hard to memorize the following proportions for future use:


For illustration in the next figure we will give you the limits of the functions tema6_EN_files\tema6_EN_MathML_55.jpg and tema6_EN_files\tema6_EN_MathML_56.jpg

Figura 1 Tema 6
Figure 1

Figura 2 Tema 6
Figure 2

If the function tema6_EN_files\tema6_EN_MathML_57.jpg is defined and continuous in the interval tema6_EN_files\tema6_EN_MathML_58.jpg, then for every point tema6_EN_files\tema6_EN_MathML_59.jpgbelonging to this interval it can be written down that

tema6_EN_files\tema6_EN_MathML_60.jpg (1.1) 

or more figuratively put, if after the substitution of tema6_EN_files\tema6_EN_MathML_61.jpg with tema6_EN_files\tema6_EN_MathML_62.jpg the result is a finite number then that is the sought limit. However this is unlikely to happen to you in a serious exam. You would be required to do something more. You will see later on.

Example 1.3. Find the limits of the function tema6_EN_files\tema6_EN_MathML_63.jpg when tema6_EN_files\tema6_EN_MathML_64.jpgand tema6_EN_files\tema6_EN_MathML_65.jpg, and the limit of the function tema6_EN_files\tema6_EN_MathML_66.jpg when tema6_EN_files\tema6_EN_MathML_67.jpg.

Solution: Here formula (1.1) can be used for both functions:


2  Rules for seeking limits. Examples.

tema6_EN_files\tema6_EN_MathML_71.jpg (2.1) 
tema6_EN_files\tema6_EN_MathML_72.jpg (2.2) 

In particular tema6_EN_files\tema6_EN_MathML_73.jpg

tema6_EN_files\tema6_EN_MathML_74.jpg (2.3) 
tema6_EN_files\tema6_EN_MathML_75.jpg (2.4) 
tema6_EN_files\tema6_EN_MathML_76.jpg (2.5) 
tema6_EN_files\tema6_EN_MathML_77.jpg (2.6) 

We continue by examining an interesting example.

Example 2.1. Find the limit of the function tema6_EN_files\tema6_EN_MathML_78.jpgwhen tema6_EN_files\tema6_EN_MathML_79.jpg.

Solution: Here we apply the rules (2.1), (2.2) and (2.3):


The result is the so called indefinite form of the kind tema6_EN_files\tema6_EN_MathML_81.jpg. This means that we still can't determine the limit of the given function.

Now we will apply a different technique to this example:


Except the shown indefiniteness there are other indefinite types


In such cases transformations are made in the analytical expression of the function and rules for limit seeking are only then applied.

Such examples should be expected at a serious exam.

Definition 2.2. Finding a limit in case of indefiniteness when such a limit exists is called indefiniteness expansion.

A useful rule

If we have to determine the limit of a function of the form


then its best that in the numerator and in the denominator we put as multiplier before brackets the highest exponent of tema6_EN_files\tema6_EN_MathML_85.jpg. In that case the given function tema6_EN_files\tema6_EN_MathML_86.jpg would look like this


Example 2.3. When tema6_EN_files\tema6_EN_MathML_88.jpgfind the limits of the functions




3  Limits of some basic functions. Illustrations.


tema6_EN_files\tema6_EN_MathML_94.jpg (3.1) 
Figura 3 Tema 6
tema6_EN_files\tema6_EN_MathML_95.jpg (3.2) 
Figura 4 Tema 6
tema6_EN_files\tema6_EN_MathML_96.jpg (3.3) 

Figura 5 Tema 6
tema6_EN_files\tema6_EN_MathML_97.jpg (3.4) 
Figura 6 Tema 6

In particular


4  A few examples and their solutions

Example 4.1. Find the limit tema6_EN_files\tema6_EN_MathML_100.jpg

Solution: After substituting tema6_EN_files\tema6_EN_MathML_101.jpg with the limit value of tema6_EN_files\tema6_EN_MathML_102.jpg we get an indefiniteness of the type tema6_EN_files\tema6_EN_MathML_103.jpg To cope with the indefiniteness we put as a multiplier before brackets the highest exponent of tema6_EN_files\tema6_EN_MathML_104.jpg both in the nominator and the denominator:


Example 4.2. Find the limit tema6_EN_files\tema6_EN_MathML_106.jpg

Solution: Since tema6_EN_files\tema6_EN_MathML_107.jpg we have an indefiniteness of the type tema6_EN_files\tema6_EN_MathML_108.jpg. (Sometimes instead of tema6_EN_files\tema6_EN_MathML_109.jpgor tema6_EN_files\tema6_EN_MathML_110.jpg is used tema6_EN_files\tema6_EN_MathML_111.jpg or tema6_EN_files\tema6_EN_MathML_112.jpg)


We replace tema6_EN_files\tema6_EN_MathML_114.jpg. Then for tema6_EN_files\tema6_EN_MathML_115.jpgwe have tema6_EN_files\tema6_EN_MathML_116.jpg (or tema6_EN_files\tema6_EN_MathML_117.jpg). Consequently tema6_EN_files\tema6_EN_MathML_118.jpg

Example 4.3. Find the limit tema6_EN_files\tema6_EN_MathML_119.jpg

Solution: Here the indefiniteness is of the kind tema6_EN_files\tema6_EN_MathML_120.jpg, because tema6_EN_files\tema6_EN_MathML_121.jpgWe have


Example 4.4. Find the limit tema6_EN_files\tema6_EN_MathML_123.jpg

Solution: After substituting tema6_EN_files\tema6_EN_MathML_124.jpg with the limit value tema6_EN_files\tema6_EN_MathML_125.jpg the result is an indefiniteness of the type tema6_EN_files\tema6_EN_MathML_126.jpg. We will try to expand it:


Example 4.5. Find the limit tema6_EN_files\tema6_EN_MathML_129.jpg

Solution: Again the indefiniteness at hand is of the type tema6_EN_files\tema6_EN_MathML_130.jpg. We will use the formula tema6_EN_files\tema6_EN_MathML_131.jpg

If we replace tema6_EN_files\tema6_EN_MathML_132.jpg and tema6_EN_files\tema6_EN_MathML_133.jpg the formula above will look like this




Example 4.6. Find the limit tema6_EN_files\tema6_EN_MathML_138.jpg

Solution After substituting tema6_EN_files\tema6_EN_MathML_139.jpg with tema6_EN_files\tema6_EN_MathML_140.jpg or tema6_EN_files\tema6_EN_MathML_141.jpg the result is respectively an indefiniteness of the type tema6_EN_files\tema6_EN_MathML_142.jpgor tema6_EN_files\tema6_EN_MathML_143.jpg.


Example 4.7. Find the limit tema6_EN_files\tema6_EN_MathML_145.jpg

Solution: The indefiniteness of the example is of the type tema6_EN_files\tema6_EN_MathML_146.jpgWe have


After the transformations the indefiniteness is still tema6_EN_files\tema6_EN_MathML_148.jpg. Probably gained experience tells you that both in nominator and denominator we need to put before brackets as a multiplier the respective highest exponent of tema6_EN_files\tema6_EN_MathML_149.jpg:


Example 4.8. Find the limit tema6_EN_files\tema6_EN_MathML_151.jpg

Solution: We need to use the formula tema6_EN_files\tema6_EN_MathML_152.jpgThen tema6_EN_files\tema6_EN_MathML_153.jpg.


Example 4.9. Find the limittema6_EN_files\tema6_EN_MathML_155.jpg

Solution: After the substitution of tema6_EN_files\tema6_EN_MathML_156.jpg with the limit value (in the case tema6_EN_files\tema6_EN_MathML_157.jpg) the result is an indefiniteness of the type tema6_EN_files\tema6_EN_MathML_158.jpg.


Problems for individual work

Problem 4.10. Below are the graphics of for functions tema6_EN_files\tema6_EN_MathML_160.jpg and tema6_EN_files\tema6_EN_MathML_161.jpg.You have to determine what are the limits of the first three functions when tema6_EN_files\tema6_EN_MathML_162.jpgand the limits of the forth when tema6_EN_files\tema6_EN_MathML_163.jpgtema6_EN_files\tema6_EN_MathML_164.jpg, tema6_EN_files\tema6_EN_MathML_165.jpg.

Figura 7 Tema 6

Figura 8 Tema 6
Figura 9 Tema 6
Figura 10 Tema 6

Exercise 4.11. Below is the graphics for the function tema6_EN_files\tema6_EN_MathML_166.jpg, which is defined in intervals tema6_EN_files\tema6_EN_MathML_167.jpg.

Try to show what are the limits of the function when tema6_EN_files\tema6_EN_MathML_168.jpg and tema6_EN_files\tema6_EN_MathML_169.jpg.

Figura 11 Tema 6

Exercise 4.12. Find the limit tema6_EN_files\tema6_EN_MathML_170.jpg.

Answer tema6_EN_files\tema6_EN_MathML_171.jpg.

Exercise 4.13. Find the limit tema6_EN_files\tema6_EN_MathML_172.jpg.

Answer tema6_EN_files\tema6_EN_MathML_173.jpg.

Exercise 4.14. Find the limit tema6_EN_files\tema6_EN_MathML_174.jpg

Answer tema6_EN_files\tema6_EN_MathML_175.jpg

Exercise 4.15. Find the limit tema6_EN_files\tema6_EN_MathML_176.jpg.

Answer tema6_EN_files\tema6_EN_MathML_177.jpg.

Conclusion If you have in the end understood what has been read up to this point you wouldn't be surprised when you encounter something written down this way tema6_EN_files\tema6_EN_MathML_178.jpg, which means that tema6_EN_files\tema6_EN_MathML_179.jpg or tema6_EN_files\tema6_EN_MathML_180.jpg, which is the same as tema6_EN_files\tema6_EN_MathML_181.jpg Interpretation is done analogically for tema6_EN_files\tema6_EN_MathML_182.jpg, tema6_EN_files\tema6_EN_MathML_183.jpg, tema6_EN_files\tema6_EN_MathML_184.jpg etc.

By Ilia Makrelov, Plovdiv university,