EVALUATING LIMITS OF EXPONENTIAL FUNCTIONS

INTRODUCTION

An exponential function is used to measure rapid change in a quantity. This quantity is known as the base and the rate at which it is increasing or decreasing is known as its exponential function. Let’s take ‘a’ as a variable. When ‘a’ exponentially increases, it is called exponential growth. Similarly, when ‘a’ decreases, it is called exponential decay. The growth or decay is represented by exponential functions. For example, ax  is an exponential function where ‘a’ is a real number and is called the base. ‘x’ is an independent variable with a range on the positive side of the number line.

Limits are values that are approached by variables of functions. Because this exponential function is both dynamic and continuous, evaluating limits becomes necessary to find the value of a function at different stages of its change. The laws of limits help in calculation of enormous data or hypothetical situations, such as surfaces of other planets. Limits and continuity are interlinked. Continuity means that a function is growing or decaying evenly. Limits are applicable to all continuous functions.

Types of LIMITS

To learn the process of evaluating limits, we must understand the two different directions via which an exponential function can approach a limit. These can be considered as types of limits. They are as follows;

• RIGHT-HAND LIMIT

When the variable approaches a limit(the constant) by increasing in value, it is the case of exponential growth. Thus, the limit is being approached from a smaller value. In such cases, the limit falls to the right-hand side of the variable on the number line. Such a limit is called a right-hand limit. This can be written in the equation form as:

Lim x→a

f(x)= L+

It means that the function ‘x’ approaches the limit ‘L’ from the right hand side.

• LEFT-HAND LIMIT

When the variable approaches a limit(the constant) by decreasing in value, it is the case of exponential decay. Radioactivity can be considered an example of exponential decay. Here, the limit is approached from a larger value. Hence, the limit falls to the left-hand side of the variable on the number line. Such a limit is called a left hand limit. It can be written as;

Lim x→a

f(x)=L-

It shows that the function ‘x’ approaches the limit ‘L’ from the left hand side.

Laws of Limits

The properties of limits help in the evaluation of limits. While working with multiple variables, the addition, subtraction, multiplication and division of limits are essential. They help us to determine values without knowing the values of variables. This can only be done through limits. It must be kept in mind that these calculations are different from that of integers. There are certain exceptions to the laws of limits but generally, they are as follows:

• Law of Constants– The rule says that the limit of a constant variable is also constant. This observation can be made easily through a graph. In the equation form it will look like,

Lim c=c

a→x

• Law of addition – To determine the limit of the sum of two variables, you must simply add the values of their limits. 

It will be written as an exponential equation;

 lim [ f(x) + g(x) ] =Lim f(x)+ Lim g(x)

  x→a                   x→a        x→a 

• Law of subtraction – The difference of two variables can be found by subtracting its limits. 

It will be represented as;

Lim[ (f(x)-g(x) ] = Lim f(x)- Lim g(x)

      x→a              x→a       x→a

• Law of multiplication– The limit of the product of two variables is the product of its limits.

It will be shown as;

Lim [ f(x).g(x) ] = Lim f(x).g(x) 

       x→a              x→a       x→a

• Law of Division– The limit of the quotient of two variables is the quotient of its limits.

It can be understood by;

Lim [ f(x)/g(x) ] = Lim f(x)/ Lim g(x) 

       x→a             x→a       x→a

• Law of Roots– The limit of the root of two variables is the root of their limits.

In an equation;

Lim n ⇃f(x)= n ⇃Lim f(x)

                                              x→a              x→a 

It is to be noted that ‘n’ is an integer in this equation.

• Law of Powers– The limit of the exponent of two variables is the exponent of the limit of the variable, i.e., 

Lim f(x)n = (Lim f(x))n

                        x→a          x→a 

It is to be noted that ‘n’ is an integer in this equation.

LIMITS and CONTINUITY

Limits and continuity are interlinked. Continuity means that a function is growing or decaying evenly. Limits are applicable to all continuous functions. The continuity of functions can be observed on graphs as well. They will not have any breaks or bumps. All functions that do not follow these three criteria are discontinuous;

• The relation f(a) should exist.
• The function f(x) should have a limit as x approaches ‘a’.
• The limit of f(x) as  x→a must be equal to f(a)

CONCLUSION

The evaluation of limits of exponential functions is crucial to calculus. They are applicable to many real life problems. The concept is used in understanding the behaviour of variables in dynamic systems. Differentiation and Integration are incomplete without the evaluation of limits. The change in variables can be studied by the laws of limits. Exponential functions can be broken down into equations and quantified. The concept of limits is closely interrelated to continuity. To understand continuity, limits are the key. Limits are applicable in various fields like astronomy, data sciences, biotechnology, quantum physics, oceanology, etc. They are indispensable in the subject of mathematics and technology studies.