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VSAT 2023

Introduction of Limits

Last updated date: 23rd Jan 2023
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A limit is a value at which a function approaches the output for the given input values in mathematics. Limits are used to define integrals, derivatives, and continuity in calculus and mathematical analysis. It's employed in the analysis process, and it's always concerned with the function's behavior at a specific time. The concept of the limit of a topological net is connected to the limit and direct limit in the theory category, and the limit of a sequence is further generalized in the concept of the limit of an earth science net. Integrals are generally divided into two categories: definite and indefinite integrals. The upper and lower limits of definite integrals are properly defined.

What is The Limits Definition?

A limit is a value that a function or sequence approaches as the input approaches a certain value in mathematics. Limits are used to define continuity, derivatives, and integrals, which are all important topics in calculus and mathematical analysis. The concept of a sequence limit is expanded to include the concept of a topological net limit, and it is strongly related to the concepts of limit and direct limit in category theory.


A function's limit is commonly stated in formulas.


$\displaystyle \lim_{x \to c}f(x)= L$


and is read as "the limit of ‘f’ of ‘x’ as x approaches to c equals L". A right arrow (→) denotes the fact that a function f approaches the limit L as x approaches c.


$f(x)\to L$ as $x \to c$


Limit of a Function

Consider the case where f is a real-valued function and c is a real number. Intuitively speaking, the expression,


$\displaystyle \lim_{x \to c}f(x)= L$


means that by bringing x close enough to c, f(x) can be made to be as close to L as required. In that case, the previous equation might be read as "when x approaches c, the limit of f of x is L."


In 1821, Augustin-Louis Cauchy and Karl Weierstrass codified the definition of a function's limit, which became known as the (, ) definition of limit. (The lowercase Greek letter epsilon (ε)) is used in the definition to indicate any small positive number, so that "f(x) becomes very close to L" means that f(x) eventually lies in between (L − ε, L + ε), which can also be written using the absolute value sign as |f(x) − L| < ε.


The phrase "as x approaches c (x→c)" then indicates that we refer to values of x, whose distance from c is less than some positive number δ (the lower case Greek letter delta (δ)) that is, values of x within either (c − δ, c) or (c, c + δ), which can be expressed with 0 < |x − c| < δ. The 1st inequality means that the distance between x and c is greater than 0 and that x ≠ c, while the second indicates that x is within distance δ of c.


Even if f(c) ≠ L, the above definition of a limit is true. The function f does not even have to be defined at c.


Properties of Limit

A list of limit properties is provided below:

Assume that $\displaystyle \lim_{x \to a} f(x)$ and $\displaystyle \lim_{x \to a} g(x)$ exist and c is a constant. Then,

1. $\displaystyle \lim_{x \to a}\left [ c.f(x) \right ]= c \displaystyle \lim_{x \to a} f(x)$

The factor of a constant is multiplicative out of a limit.

$\displaystyle \lim_{x \to a}\left [ f(x) \right ]= c \displaystyle \lim_{x \to a} f(x)$


2. $\displaystyle \lim_{x \to a}\left [ f(x) \pm g(x)\right ]= \displaystyle \lim_{x \to a} f(x) \pm \displaystyle \lim_{x \to a} g(x)$

Select the limits individually and replace them with the corresponding sign to consider the limit of a sum of differences. This fact works anyway of several functions we separated by “+” or “-”.


3. $\displaystyle \lim_{x \to a}\left [ f(x) . g(x)\right ]= \displaystyle \lim_{x \to a} f(x) . \displaystyle \lim_{x \to a} g(x)$

Consider product limits in the same way that sums and differences have limits. Simply select the piece limit and reassemble, and this is not restricted to only two functions.


4. $\displaystyle \lim_{ x\to a}\left [ \frac{f(x)}{g(x)} \right]=\frac{\displaystyle \lim_{x \to a}f(x)}{\displaystyle \lim_{x \to a}g(x)}$

Provided $\displaystyle \lim_{ x\to a}g(x)\neq 0$

As you can see, we only need to care about the quotient limit if the denominator's limit is zero. It would result in a division by zero error if it were zero.


5. $\displaystyle \lim_{ x\to a}c= c$

The limit of every constant is only a constant. By visualizing the graph of the function f(x) = c, you can easily understand it.


6. $\displaystyle \lim_{ x\to a} xn = an$


Solved Examples

Example 1: To compute $\displaystyle \lim_{x \to 5}\left ( 6x^{2}+2x-4 \right )$

Solution:

To compute, divide the limit into three parts using property 2.

$\displaystyle \lim_{x \to 5}\left ( 6x^{2}+2x-4 \right )= \displaystyle \lim_{x \to 5}\left ( 6x^{2} \right )+ \displaystyle \lim_{x \to 5}\left ( 2x \right )-\displaystyle \lim_{x \to 5}4$

= 6(25) + 2(5) - 4

= 150 + 10 - 4

= 156


Example 2: To compute $\displaystyle \lim_{ x\to 3}\left [ \frac{(x-1)}{(x-2)} \right ]$

Solution:

By using the fourth property of limit divide the limit in numerator and denominator.

$\displaystyle \lim_{ x\to 3}\left [ \frac{(x-1)}{(x-2)} \right ]=\frac{\displaystyle \lim_{x \to 3}(x-1)}{\displaystyle \lim_{x \to 3}(x-2)}$

$ = \frac{(3-1)}{(3-2)}$

$=\frac{2}{1}$

= 2


Example 3: To compute $\displaystyle \lim_{x \to -2}\left ( 12x^{2}+4x-16 \right )$

Solution:

To compute, divide the limit into three parts using property 2.

[\displaystyle \lim_{x \to -2}\left ( 12x^{2}+4x-16 \right )= \displaystyle \lim_{x \to -2}\left ( 12x^{2} \right )+ \displaystyle \lim_{x \to -2}\left ( 4x \right )-\displaystyle \lim_{x \to -2}(16)]

$ =12(-2)^{2}+4(-2)-16$

= 12(4) + 4(-2) - 16

= 48 - 8 -16

= 24


Practise Question MCQs

1. The value of the limit $\displaystyle \lim_{ x\to 0}(cosx)^{cot2x}$ is

  1. 1

  2. e

  3. e1/2

  4. e

Ans: d


2. The value of limit $\displaystyle \lim_{ x\to 0}\frac{{sin (a+x)-sin (a-x)}}{x}$ is

  1. 0

  2. 1

  3. 2 cosa

  4. 2 sina

Ans: C


Conclusion

A limit is a value at which a function approaches the output for the given input values in mathematics. In calculus and mathematical analysis, limits are used to define integrals, derivatives, and continuity. We looked at limt functions in this article and solved questions to help us understand them better.

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FAQs on Limits

1. What is the class limit?

The minimum and maximum values that the class interval can include can be defined as the class limits. The lower class limit (LCL) is the lowest value, and the upper-class limit (UCL) is the highest value (UCL).

2. What is the limit meaning in the English language?

The limit meaning in the English language is, limit is the maximum amount, extent, or degree of something that can be achieved. A specific form of limit is the maximum or minimum amount of something, such as time or money, that is permitted by a rule, legislation, or decision.

3. What are limits used for in real life?

Limits are also used to approximate the calculation of derivatives in real life. In real-life situations, computing a derivative of intricate motions is quite difficult.

4. Who invented limits?

Archimedes of Syracuse.


In the third century B.C., Archimedes of Syracuse invented the concept of limits to measure curved figures and the volume of a sphere. The needed quantity can be obtained by carving these figures into little parts that can be approximated, and then increasing the number of pieces.

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