Limit is an important concept of calculus that specifies the value the function approaches as the input of the function gets closer and closer to a specific number. This concept is the foundation of calculus. In this article, we will discuss the limit rules or laws of the limit with examples. But before proceeding to these rules, first, we will discuss the concept of limit in detail.

Limit - Introduction

The limit in mathematics is defined as a value that is approached by the function's output for the given values of input. The concept of limit is very important in mathematics and calculus as it helps to define integrals, continuity and derivatives. This concept is also used to analyze a process because it tells us the behavior of the function at a specific point. Integration is an important concept of calculus. We all know that there are two types of integrals known as definite integrals and indefinite integrals. The definite integrals have upper and lower limits that are defined properly. On the other hand, upper and lower limits do not exist in indefinite integrals.

In the next section, you will find the formal definition of the limit along with the mathematical notation.

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Limit - Definition

Limits  tell us how a function performs when its independent variable, i.e.,  x, approaches a specific value. The proper definition of limit is given below:

Consider a function f that is defined on a specific open interval that has a number "a", except at "a" itself. Thus, we can say that the limit of f(x) as x approaches to "a" is L.  The mathematical notation of this statement is given below:

    \[\lim_ {x\rightarrow a} f(x) = L\]

Consider that for every number \varepsilon >0, there is a corresponding number \delta > 0 in such as way that:

|f(x) - L| < \varepsilon

whenever:

0 < |x - a| < \varepsilon

 

Now, we will discuss some of the rules of limit along with the relevant examples.

Limit Sum Rule

The limit sum rule states that the limit of the sum is equal to the sum of the limits. Mathematically, we can write this rule as:

    \[\lim_{x \rightarrow a} [f(x) + g(x)] = \lim_{x \rightarrow a} f(x) + \lim_{x \rightarrow a} g(x)\]

Example

Evaluate the function

    \[\lim_{x \rightarrow 2} 2x^2 + 3x\]

Solution

We will use the limit sum rule here as there is a positive sign between the two terms of the function.

    \[= \lim_{x \rightarrow 2} 2x ^2 + \lim_{x \rightarrow 2} 3x\]

= 2 (2)^2 + 3 (2)

= 8 + 6

= 14

 

Limit Difference Rule

The limit difference rule states that the limit of the difference is equal to the difference of the limits. Mathematically, we can write this rule as:

    \[\lim_{x \rightarrow a} [f(x) - g(x)] = \lim_{x \rightarrow a} f(x) - \lim_{x \rightarrow a} g(x)\]

Example

Evaluate the function

    \[\lim_{x \rightarrow -1} 5x^2 - 7x\]

Solution

We will use the limit difference rule here as there is a negative sign between the two terms of the function.

    \[= \lim_{x \rightarrow -1} 5x ^2 - \lim_{x \rightarrow -1} 7x\]

= 5 (-1)^2 - 7 (-1)

= 5 + 7

= 12

 

Limit Constant Rule

This rule states that the limit of the constant function is equal to the constant. Mathematically, we can write it as:

    \[\lim_{x \rightarrow a} k = k\]

Example

Evaluate the function

    \[\lim_{x \rightarrow a} 12\]

.

Solution

The limit of the constant is equal to the constant itself. So, the limit of this function is equal to 12.

 

Constant Multiple Rule

This rule states that the limit of constant multiplied by the function is equal to the product of constant and the limit of the function. Mathematically, we can write this rule like this:

    \[\lim_{x \rightarrow a} c f(x) = c \lim_{x \rightarrow a}f(x)\]

Example

Evaluate the limit

    \[\lim_{x \rightarrow -1} 3 (x^3 + x^2 + x)\]

Solution

Here, we will first take limit of the function and then multiply the obtained value with constant term. We will find the limit of the function using limit sum rule because there is a positive sign between three terms.

    \[=3 \cdot \lim_{x \rightarrow -1} x^3 + \lim_{x \rightarrow -1} x^2 + \lim_{x \rightarrow -1} x\]

= 3 ( (-1)^3 + (-1)^2 + (-1))

= 3 (-1 + 1 - 1)

= 3(-1)

= -3

 

Limit Product Rule

This rule says that the limit of the product of two functions is equal to the product of the limits of two functions. Mathematically, we can write this rule as:

    \[\lim_{x \rightarrow a} [ f(x) g (x)] = \lim_{x \rightarrow a} f(x) \cdot  \lim_{x \rightarrow a} g(x)\]

Example

Evaluate the function

    \[\lim_{x \rightarrow 2} 4x sin x^3\]

.

Solution

In this example, we will employ the limit product rule because two separate functions are multiplied with each other. The first function is 4x and the second one is sin x^3.

    \[=\lim_{x \rightarrow 2} 4x \cdot \lim_{x \rightarrow 10} sin x^3\]

= 4 (2) \cdot sin (2)^3

= 8 \cdot sin (8)

= 8 \cdot 0.989

= 7.914

 

Limit Quotient Rule

This rule states that the limit of the quotient is equal to the quotient of the limits. Mathematically, we can denote this rule like this:

    \[\lim_ {x \rightarrow a} \frac {f (x) } {g(x)} = \frac{\lim_ {x \rightarrow a} f(x)} {\lim_ {x \rightarrow a} g(x)}}\]

Example

Evaluate the function

    \[\lim_ {x \rightarrow 2} \frac {6x + 5}{x^2 + 3x}\]

Solution

We will use the limit quotient rule here. To use this rule, first, we will apply the limits separately to the numerator and the denominator and then take the quotient.

    \[= \frac{\lim_ {x \rightarrow 2} 6x + 5} {\lim_ {x \rightarrow 2} x^2 + 3x}}\]

= \frac {6 (2) + 5} {(2)^2 + 3(2)}

= \frac{17}{10}

=1.7

 

Limit Power Rule

This rule states that the limit of the function involving a power is equal to the limit of the function raised to the power. Mathematically, we can denote this rule as shown below:

    \[\lim_ {x \rightarrow a} [f(x)]^n = [\lim_ {x \rightarrow a} f(x)]^n\]

Example

Evaluate the function

    \[\lim_ {x \rightarrow 1} [x^2 + 8x]^2\]

.

Solution

Here, we will use the limit power rule.

    \[=[\lim_ {x \rightarrow 1} x^2 + 8x]^2\]

= [(1)^2 + 8(1)]^2

= (1 + 8)^2

= (9)^2

= 81

 

Limit of a Square Root Function

This rule states that the limit of the square root function is equal to the square root of the limit of the radicand. The same rule applies for the higher roots. Mathematically, we denote this rule like this:

    \[\lim_ {x \rightarrow a} \sqrt {f(x)} = \sqrt { \lim_ {x \rightarrow a} f(x)}\]

Example 1

Evaluate the function

    \[\lim_ {x \rightarrow 2} \sqrt {3x^2 + 9x}\]

.

Solution

We can write this function as:

    \[= \sqrt { \lim_ {x \rightarrow 2} 3x^2 + 9x}\]

= \sqrt {3 (2)^2 + 9 (2)}

= \sqrt {12 + 18}

= \sqrt{30}

= 5.477

 

Example 2

Evaluate the function

    \[\lim_ {x \rightarrow 3} \sqrt [3] {9x^2 + x}\]

.

Solution

We can write this function as:

    \[= \sqrt [3] { \lim_ {x \rightarrow 2} 9x^2 + x}\]

= \sqrt [3][ {9 (3)^2 + (3)}

= \sqrt[3] {81 + 3}

= \sqrt[3]{84}

= 4.76

 

 

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Rafia Shabbir