The concept of limits is a vital basis of calculus and mathematical analysis, forming the basis for derivative and integral calculus. In mathematics, the concept of limits is used to describe the behavior of a function as its input approaches a particular value.
They provide a main concept for studying the behavior of functions, determining convergence and divergence, and solving a wide range of mathematical problems. In the 20th century, mathematicians such as Henri Lebesgue and Emile Borel made important contributions to the theory of limits, particularly in the context of measure theory.
Limits have found the most common application in different fields, including physics, engineering, economics, and computer sciences. In this article, we will discuss the basic definition of Limits, the mathematical form of Limits, the different types of Limits, and how to calculate limits in detail.
Table of Contents
What is Limit Calculus?
In mathematics, the concept of limits is used to describe the behavior of a function as its input approaches a particular value. It allows us to understand how a function behaves near a specific point or as the input values get arbitrarily close to a certain value.
Limits Formula:
Commonly, g(z) be a function defined on some interval about a point a, but possibly at itself. The limit of g(z) as z approaches a is represented as:
Lim(z→a) g(z) = M
This means that as z becomes randomly close to a and corresponding values of g(z) approach the value M. In other words, M denotes the value that the function “approaches” or becomes arbitrarily close to as z gets closer to a.
Different Types of Limit Calculus
Here are different types of limits in mathematics given below.
One-Sided Limits:
These limits describe the behavior of a function as the input approaches a specific value from either the left or the right side. One-sided limits are denoted as Lim(x→-a) f(x) and lim(x→+a) f(x) and represent the limit as “x” approaches “a” from the left and right sides respectively.
Two-Sided Limits:
Two-sided limits are called standard limits. This type of limit describes the overall behavior of a function as the input approaches a particular value. It considers both the left and right sides of the point. The two-sided limit is denoted as lim(x→a) f(x) and is equal to “L” if and only if both the left and right limits converge to “L”.
Infinite Limits:
An infinite limit occurs when the values of a function become arbitrarily large as the input approaches a certain value. If the function grows without bound as the input approaches a”. we say that the limit is infinite. This is represented as lim(x→a) f(x) = ±∞.
Limits at Infinity:
These limits describe the behavior of a function as the input becomes arbitrarily large or small. If the function approaches a specific value “L” as “x” approaches positive infinity or negative infinity and the limit is denoted as lim(x→±∞) f(x) = L.
Limit at a Point:
This mentions the limit of a function at a specific point denoted as lim(x→a) f(x). It describes the behavior of the function as the input gets arbitrarily close to the point “a” regardless of the actual value of the function itself.
How to Calculate Limit Calculus Problems?
Here is guidance on how to calculate limits.
- Determine the limit that has to be evaluated. Determine the value at which the input is approaching and the function involved.
- Substitute the value at which the input is approaching into the function. If direct substitution yields a defined value, you have found the limit. If it leads to an indeterminate form proceed to the next step.
- Use factoring, canceling common factors, or simplifying the numerator or denominator to eliminate indeterminate forms such as 0/0 or ∞/∞.
- Apply known limit theorems. Use the sum/difference rule, product rule, quotient rule, power rule, or other relevant theorems to simplify the expression and evaluate the limit.
- If the limit still cannot be determined check if the function is bounded between two known functions. Apply the squeeze theorem to establish the limit based on the bounding functions.
- If you encounter an indeterminate form such as 0/0 or ∞/∞ consider using L’Hopital’s Rule. Differentiate the numerator and denominator, and then evaluate the limit of the derivatives. Repeat this process if necessary.
- If the function involves trigonometric functions use trigonometric identities to simplify the expression and evaluate the limit.
- Employ Taylor series expansions for functions that can be represented as a series. Use the appropriate terms of the series to approximate the limit.
- If the limit involves logarithmic functions use logarithmic properties to simplify the expression and evaluate the limit.
- Repeat the above steps as needed until you can determine the limit value or conclude that the limit does not exist.
Evaluation of Limits:
Here are some examples to calculate limits problems.
Example 1:
Evaluate the following Limx→5(4x4 – 3x3 + x2 + 4).
Solution
Step 1:
First of all, we write the given functions.
Limx→5(4x4 – 3x3 + x2 + 4).
Step 2:
apply the limits individually and the coefficient separately.
Limx→5(4x4 – 3x3 + x2 + 4) = 4Limx→5(x4) – 3Limx→5(x3) + Limx→5(x2) + Limx→5(4)
Step 3:
Put the limits x→5.
Limx→5(4x4 – 3x3 + x2 + 4) = 4 (5)4 – 3(5)3 + (5)2 + 4
Step 4:
Simplify the above value.
Limx→5(4x4 – 3x3 + x2 + 4) = 4×625 – 3×125 + 25 + 4
Limx→5(4x4 – 3x3 + x2 + 4) = 2500 – 375 + 25 + 4
Limx→5(4x4 – 3x3 + x2 + 4) = 2529 – 375
Limx→5(4x4 – 3x3 + x2 + 4) = 2154
Step 5:
Therefore, Limx→5(4x4 – 3x3 + x2 + 4) = 2154.
Example 2:
Calculate the given Limx→3(15x7 + 7x5 + 3x2 – 4x).
Solution
Step 1:
First of all, we write the given functions.
Limx→3(15x7 + 7x5 + 3x2 – 4x)
Step 2:
apply the limits individually and the coefficient separately.
Limx→3(15x7 + 7x5 + 3x2 – 4x) = 15Limx→3(x7) + 7Limx→3(x5) + 3Limx→3(x2) – 4Limx→3(x)
Step 3:
Put the limits x→3.
Limx→3(15x7 + 7x5 + 3x2 – 4x) = 15 (3)7 + 7(3)5 + 3(3)2 – 4(3)
Step 4:
Simplify the above value.
Limx→3(15x7 + 7x5 + 3x2 – 4x) = 15×2187 + 7×243 +3×9 – 12
Limx→3(15x7 + 7x5 + 3x2 – 4x) = 32805 + 1701 + 27 – 12
Limx→3(15x7 + 7x5 + 3x2 – 4x) = 34533 – 12
Limx→3(15x7 + 7x5 + 3x2 – 4x) = 34521
Step 5:
Hence, Limx→3(15x7 + 7x5 + 3x2 – 4x) = 34521.
Conclusion:
In this article, we have discussed the basic definition of Limits, the mathematical form of Limits, the types of Limits, and how to calculate Limits in detail. Also, Limits explains with the help of examples. And you can easily understand Limits and solve related problems.
