LIMIT AND CONTINUITY

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. limx→2f(x)=8

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

• f(k) is defined.
• limx→kf(x)exists.
• \[\lim_{x\right arrow k^{+}} f(x) = \lim_{x\right arrow 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,

limx→0f(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 limx→k+f(x)=limx→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 limx→kf(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 limx→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 limx→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 limx→kf(x).

PRESENTED BY-

Lopamudra Parida( Lect. in Mathematics)