A limit is a value that a function approaches as its input gets closer to a specific point. The limit serves as the foundation of calculus that enables us to understand the behavior of functions as they approach specific values. It is essential in calculus and mathematical analysis for defining integrals, derivatives, and continuity. In this article, we will explore the definition of limit calculus with its mathematical representation. We will examine common types, properties, and many examples of limit calculus. Let’s start this amazing journey of mathematics with the definition of limit.
Definition of Limit
A limit represents the value a function approaches as its input approaches a specific point. Mathematically, it is expressed as:
Where x is the input variable, c is the specific point, f(x) is the function, and L is the limit value. Limits provide insights into the behavior of functions near a certain point without necessarily reaching that point.
Types of Limit
Different common types of limits are given below:
- One-Sided Limit: This examines the behavior of the function as the input gets close to a certain value from the left (lim x → c- f (x) or right side (lim x → c- f (x).
- Two-Sided Limit: It focuses on how a function acts as the input approaches a value from both the left and right sides simultaneously.
- Infinite Limit: This occurs when the value of the function becomes extremely large (positive or negative infinity) as the input gets close to a certain value. It can be expressed mathematically as
- Lim x → c f(x) = ± ∞.
- Limit of Infinity: Describes how a function behaves as the input values become extremely large or extremely small. Mathematically, it can be represented as lim x → ±∞ f(x).
Properties of Limit
If Lim x→ c f(x), Lim x→ c g(x) exist and k is any constant then the following properties hold:
- Sum/Difference Rule
Lim x→ c [f(x) ± g(x)] = Lim x→ c f(x) ± Lim x→ c g(x)
The limit of the sum or difference of two functions is equal to adding or subtracting the limits of the individual functions.
- Product Rule
Lim x→ c [f(x). g(x)] = Lim x→ c f(x). Lim x→ c g(x)
The limit of the product of two functions is equal to the product of their limits.
- Quotient Rule
Lim x→ c (f(x) / g(x)) = Lim x→ c f(x) / Lim x→ c g(x), where Lim x→ c g(x) ≠ 0
The limit of the quotient of two functions is equivalent to the quotient of their limits, provided the limit of the denominator is not zero.
- Constant Multiple Rule
Lim x→ c (k. f(x)) = k. Lim x→ c (f(x))
If a constant is multiplied by a function then the limit of the function equals the constant multiplied by the constant.
- Power Rule
Lim x→ c (f(x)) k = (Lim x→ c f(x)) k The limit of a power of a function is equivalent to the power of the limit of the function.
Techniques for Evaluating Limit
Various techniques exist for evaluating limits, each catering to different types of functions and situations. Some techniques for determining the limit are given here:
- Direct Substitution: Substitute the value of the variable directly into the function if it does not result in division by zero or other undefined forms.
- Factorization: Factor the expression to simplify and possibly cancel out common terms before substituting the value.
- Rationalization: Rationalize the expression by multiplying the numerator and denominator by a suitable conjugate to eliminate radicals or complex fractions.
- L’Hôpital’s Rule: For indeterminate forms (0/0 or ∞/∞), take the derivative of the numerator and denominator successively until a determinate form is obtained.
A limit solver is the best way to find solutions to limit problems, providing step-by-step solutions using the above-mentioned methods.
Important Functions’ Limits
- Lim x→ c (xn – cn)/(x – c) = nc(n – 1) where n is an integer and c > 0
- lim n → ∞ [(1+ (1 / n)]n = e
- lim x→ 0 (ex – 1) / x = 1
- lim x→ ∞ (ex) = ∞
- lim x→ – ∞ (ex) = 0
- lim x→ ± ∞ (a / x) = 0 (where a is any real number)
- lim θ→ 0 (tan θ / θ) = 1
- lim θ → 0 Cos θ = 1
- lim θ→ 0 (Sin θ / θ) = 1
- lim θ → 0 (1 – Cos θ)/ θ = 0
Examples of Limit
Here are some examples to illustrate the concept of limits in calculus, along with their solutions.
Example 1:
Evaluate:
Lim x→ 3 (x3 + 7×2 – 4x + 7)
Solution:
Apply the Sum/Difference property:
= Lim x→ 3 (x3) + Lim x→ 3 (7×2) – Lim x→ 3 (4x) + Lim x→ 3 (7)
Use the Constant Multiple property:
= Lim x→ 3 (x3) + 7 Lim x→ 3 (x2) – 4 Lim x→ 3 (x) + 7 Lim x→ 3 (1)
Apply the Power Rule:
= [Lim x→ 3 (x)]3 + 7 [Lim x→ 3 (x)]2 – 4 Lim x→ 3 (x) + 7 Lim x→ 3 (1)
Substitute x = 3:
= (3)3 + 7 (3)2 – 4 (3) + 7
Simplify:
= 27 + 7 (9) – 12 + 7
= 27 + 63 – 12 + 7
= 85
Thus, Lim x→ 3 (x3 + 7×2 – 4x + 7) = 85
Example 2:
Compute the limit of f(x) = (x – 8) / (x2 – 64) as x approaches 8.
Solution:
Given function: f(x) = (x – 8) / (x2 – 64)
Lim x→ 8 (x – 8) / (x2 – 64) = 0/0 (That is intermediate form)
The factorization technique can be used to determine the limit of the given function.
Factor the denominator: (x2 – 64) = (x – 8) (x + 8)
∴ (a2 – b2) = (a – b) (a + b)
Lim x→ 8 (x – 8) / (x2 – 64) = Lim x→ 8 (x – 8) / (x – 8) (x + 8)
= Lim x→ 8 [1 / (x + 8)]
Apply the Quotient Rule
= (Lim x→ 8 (1)) / (Lim x→ 8 (x + 8))
Substitute x = 8
= 1 / (8 + 8) = 1/16
Therefore, the limit of f(x) = (x – 8) / (x2 – 64) as x approaches 8 is 1/16.
Conclusion
In this article, we delve into the concept of limits, complete with its mathematical representation. We discuss various types of limits and their applications and explore different techniques for evaluating them. Through carefully selected examples, we aim to enhance your understanding of this foundational calculus concept.