Limits and Continuity

Limit and Continuity Meaning

The concept of the limits and continuity is one of the most important terms to understand to do calculus. A limit is stated as a number that a function reaches as the independent variable of the function reaches a given value. For example, consider a function f(x) = 4x, we can define this as,The limit of f(x) as x reaches close by 2 is 8.


Mathematically, It is represented as \[\lim_{x\rightarrow 2} f(x) = 8 \].


A function is determined as a continuous at a specific point if the following three conditions are met.

  • f(k)  is defined.

  • \[\lim_{x\rightarrow k} f(x)\] exists.

  • \[\lim_{x\rightarrow k^{+}} f(x) = \lim_{x\rightarrow k^{-}} f(x) = f(k).


A function will only be determined as continuity of a function if its graph can be drawn without lifting the pen from the paper. But a function is defined as discontinuous when it has any gap in between.


Continuity Meaning

A function is said to be continuous if and only if it is continuous at each point of its domain. A function is determined to be continuous on an interval, or subset of its domain, if and only if it is continuous at each point of its domain. The addition, subtraction, and multiplication of continuous functions with similar domain are also continuous excluding the point in which the denominator is equivalent to zero. Continuity can also be defined with respect to limits by stating that f(x) is continuous at x₀ of its domain if and only if, for values of x in its domain,

\[\lim_{x\rightarrow 0} f(x)\] = f(x₀)


Types of Discontinuity

There are different types of discontinuity. These are,

Infinite Discontinuity

A division of discontinuity in which a vertical asymptote exists at x = a and f(k) is not defined. This is also known as asymptotic discontinuities. If a function possesses values on both sides of an asymptote, then it cannot be interlinked, so it is discontinuous at the asymptote.


Jump Discontinuity

A division of discontinuity in which \[\lim_{x\rightarrow k^{+}} f(x) = \lim_{x\rightarrow k^{-}} f(x)\] but the limit present on both sides are finite.This is also known as simple discontinuity or continuities of the first kind.


Positive Discontinuity

A division of discontinuity in which function has a predetermined two-sided limit at x = k but either f(x) is not defined at ‘k’  or its value is not equivalent to the limit at k.


Limit

A limit is defined as a value that a function reaches the output for the given set of input values. The limit of functions is important in calculus and Mathematical analysis and used to define the derivatives,integrals, and continuity.


Let us define limit by considering a real-valued function “f” and the real number “k”, the limit is usually represented as

\[\lim_{x\rightarrow k} f(x) = z\]


It is stated as “ the limit of f of x, as x approaches close to K equivalent to Z. The “ lim” represents a limit, and describes that the function f reaches the limit Z as x reaches k is determined by the right arrow.


Important Points 

  • If \[\lim_{x\rightarrow k^{-}} f(x)\] is the expected value of f at x = k stated the values of ‘f’close by x to the left side of k. This value is determined as the left-hand limit of ‘f’ at k.

  • If \[\lim_{x\rightarrow k^{+}} f(x)\] is the average value of f at x = k stated the values of ‘f’ close by x to the right side of k This value is determined as the right-hand limit of f(x) at k.

  • If the right-hand and left-hand limits meet each other, we state the common value as the limit of f(x) at x = k and represent it by \[\lim_{x\rightarrow k} f(x)\].


One - Sided Limit

The limit that relies completely on the values of a function considered at x -value that is moderately greater or less than a particular value. A two-sided limit \[\lim_{x\rightarrow k} f(x)\] considers the values of x into account that are both larger than and smaller than k. A one-sided limit from the left side \[\lim_{x\rightarrow k^{-}} f(x)\] or from the right side \[\lim_{x\rightarrow k^{+}} f(x)\] considers only values of x that are smaller or greater than k respectively.


Properties of Limit

  • The limit of a function is defined as f(x) as approaches K as x inclines to limit y, such that; \[\lim_{x\rightarrow y} f(x) = k\].

  • The limit of the addition of two functions is equivalent to the addition  of their limits, such that \[\lim_{x\rightarrow y}[f(x) + g(x)] = \lim_{x\rightarrow y}f(x) + \lim_{x\rightarrow y} g(x)]\]

  • The limit of any constant function will be the constant term, such that, \[\lim_{x\rightarrow y} C = C\].

  • The limit of multiplication of the constant and function is equivalent  to the multiplacation of constant and the limit of the function, such that: \[\lim_{x\rightarrow y}m f(x) = m \lim_{x\rightarrow y}f(x)\].

  • Quotient Rule of the limit of a function: \[\lim_{x\rightarrow y}[\frac{f(x)}{g(x)}] = \frac{\lim_{x\rightarrow y}f(x)}{\lim_{x\rightarrow y}g(x)}\], if \[\lim_{x\rightarrow y}g(x) = 0\]. 


Solved Examples

1. Evaluate the following:

\[\lim_{x\rightarrow 0} \sqrt{2 + x} - \frac{\sqrt{2}}{x}\].


Solution: Substitute y = 2 + x so that when x0 , y 0. Then,


\[\lim_{x\rightarrow 0} \sqrt{2 + x} - \frac{\sqrt{2}}{x} = y\]


= \[\frac{y^{1/2} - 2^{1/2}}{y - 2}\]


= \[\frac{1}{2} (2)^{\frac{1}{2} - 1} = \frac{1}{2} (2)^{-\frac{1}{2}} = \frac{1}{2} \sqrt{2}\]


2. Examine whether the function given below is discontinuous.


\[f(x) = \frac{x^{2} - 9}{3x^{2} + 2x - 8}\]


Solution:

While solving rational expressions in which both the numerator and denominator are continuous (as we have in the equation given above, both are polynomials) the only points in which the rational expression will be discontinuous where we get division by zero.


Hence, we just need to determine where the denominator is zero and that is quite easy for this problem.


3x² + 2x - 8 = (3x-4) (x + 2) = 0   x= 4/3 , x = -2


Hence, the function will be discontinuous at the points x= 4/3 and  x = -2.


Quiz Time

1. \[\lim_{x\rightarrow 0} \frac{\sqrt{1 - cos 2x}}{\sqrt{2x}}\] is

  1. 1

  2. -1

  3. 0

  4. Does not exist


2. A function is said to be continuous for x Є R, if

  1. It is continuous at x = 0

  2. Differentiable at x = 0

  3. Continuous at two points

  4. Differentiable for x Є R

FAQ (Frequently Asked Questions)

1. How Continuity and Limits are Closely Related to Each Other?

Limits and continuity are closely related to each other. The function can either be continuous or discontinuous. The continuity of a function states that, if there are minor variations in the input of a function then there must be minor changes in the output also. 


In elementary calculus, the condition f(x) → λ as x → k implies that the function f(x) can be formed to lie as close as we like to the number Lamba as long as we consider the number x unequal to the number ‘k’ but close enough to ‘k’ which show that f(k) might be very far from lambda and there is no requirement for f(k) to be defined. The important result we apply for the derivative of a function is f’(k) of a given function f at k can be written as 


f’(k) = limx → k f(x) - f(k) /x - k

By understanding limits, we can define the continuity in a more precise manner. With the proper understanding of the concepts of limit and continuity, you are prepared for calculus.

2. What are the Applications of Limit in Real-Life Situations?

The use of the limit of a function is wide in real-life situations. For example,

  • We can carry out chemical reactions in a beaker with two chemicals that form a new compound over time. The quantity of new compounds is the limit of a function as the time approaches close to infinity.

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  • If you put an ice cube in a glass of warm water and measure the temperature with time, you will notice that the temperature instantly approaches the room temperature. Measuring the temperature is again a limit as time approaches close to infinity.

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  • The limit of a function is also used to calculate the derivative of a function.