Understanding Limits in Calculus: Definition, Properties, & Calculation

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Limit plays an important role in calculus to determine the limiting value of the function at a specific point. Limits in calculus are used to determine the derivative and integral at the specific point / specific interval values.

In this blog, we will discuss the definition of the limit with its detailed concept, properties of the limit, and its importance in calculus. Furthermore, we will solve the different examples for a better understanding of the limit by using its properties.

Limit Definition

Limit can be defined as the value of the limiting point “x” approaches to any specific number say “c” and then the limiting value of the function “f(x)” approaches to a particular number say “L”. Mathematically, the limit of a function “f(x)” as “x” approaches to value “c” is denoted as,

Lim x→c​f(x) = L

This notation reads, as “the limit of f (x) is “L ”as“ x approaches c where “L” is the limiting value of the “f (x) that will get closer to functional value as “x” get closer to “c” Graphical representation of the limit can be represented as,

Explanation of limits

Limit in calculus is a fundamental in understanding changes in quantities at specific points. It enable us to examine the behavior of a function, how its values increase or decrease as it approaches a particular point.

Beyond offering understandings into continuity or discontinuity of a function, particularly as points near infinity, limits inspire a deeper, more complex view on mathematics. It is not only a technique; it represent a shift in how we theorize mathematical theories, setting the foundation for calculus and other concepts based on function limits.

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Critically, limits have a key role in calculus, serving as the foundation for defining derivatives (slope of the tangent line) and integrals (area under the curve). Their utility extends beyond pure mathematics, finding applications in engineering, physics, economics, and even medical models, particularly in understanding disease spread rates.

Thus, exploring limits is not just about mastering a calculus component—it is about unlocking a new dimension of thinking in mathematics and its applications across disciplines.

Properties of Limits

Limits have many important properties that make calculations easier and help to solve limits of a combination of the different functions by using various operations such as the sum, difference, product, or quotient.

Here we discuss the properties of the limit with formulas.

  • Sum/Difference Property

The limit of a sum/difference of functions is used to find the limit of the sum/difference of the two different functions by distributing them individually on a single function.

Lim x->c [f(x) ± g(x)] = Lim x->c f(x) ± Lim x->c g(x)

  • Multiplication by a Constant Property

This property helps to solve the constant multiplied by a function. The limit of constant times with a function is equal to the constant times of the limit of the function.

Lim x->c [k f(x)] = k × Lim x->c f(x)

  • Multiplication/Product Property

This property helps to solve the product of the two functions. The limit of a product of functions is equal to the product of individual limits.

Lim x->c [f(x) × g(x)] = Lim x->c f(x)  × Lim x->c g(x)

  • Division/Quotient Property
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This property helps to solve the quotient of the two functions by distributing the limit individually. The limit of a quotient of functions is equal to the quotient value of their limits. Moreover, the limiting value of the denominator is not equal to zero.

Lim x->c [f(x)/g(x)] = Lim x->c f(x)/Lim x->c g(x)

Importance of limit in calculus

Limits play an extensive role in calculus to determine the limiting solution of the different functions if the exact solution of the function does not exist. It helps to find the solution to those functions if it does not exist through direct substitution of the limiting value and also deals with infinity or undefined expression by using the special algebraic technique.

Limit enables us to find the slope of a tangent of the different functions by putting the value of limits and is also used to find the exact area under the curves no matter how the interval is too small/large.

How to calculate the limits using its properties?

Here we provide some examples to determine the limit of the function by using the properties of the limits and provide the detailed steps for a better understanding of the limit.

Example 1:

Find the limits of   “4x3 – 4” by using the properties of the limit at “x” approaches to “3”.

Solution:

Step 1: Consider the given value is equal to “f(x)”.

f(x) = 4x3 – 4

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Step 2: Apply the limit with its limiting value on both sides of the functions.

Lim x→3 f(x) = Lim x→3 (4x3 – 4)

Step 3: solve the limits by using the difference properties of the limit.

Lim x→3 f(x) = Lim x→3 (4x3 – 4)

        = Lim x→3 (4x3) – Lim x→3 (4)

Using the multiplication constant property of the limit simplify the terms.

                     = 4 (3)3 – (4)

        = 4 (27) – (4)

        = 108 – 4

Lim x→3 f(x) = 104

Example 2:

Find the limit of (x2– 5)/(x+3) using the properties of the limit if the limiting value is “2”.

Solution:

Step 1: Consider the given value is equal to “f(x)”.

f(x) = (x2– 5)/(x+3)

Step 2: Apply the limit with its limiting value on both sides of the functions.

Lim x→2 f(x) = Lim x→2 ((x2– 5)/(x+3))

Step 3: Solve the limits by using the quotient properties of the limit.

Lim x→2 f(x) = Lim x→2 ((x2– 5)/(x+3))

        = Lim x→2 (x2– 5)/ Lim x→2 (x +3)

        = ((2)2 – 5)/ (2+3)

        = (4 – 5)/ (2+3)

        = (-1/5)

Lim x→2 f(x) = – 0.2

Final Words

In this article, we’ve discussed the basic concept of limits with its definition, explanation, and the properties of the limit in detail. In addition, we discussed the importance of the limit in calculus and solved the different examples with steps for a better understanding of the limit.

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