Limits And Continuity

Limits and continuity concept is one of the most crucial topic in calculus. Both concepts have been widely explained in Class 11 and Class 12. A limit is defined as a number approached by the function as an independent function’s variable approaches a particular value. For instance, for a function f(x) = 4x, you can say that “The limit of f(x) as x approaches 2 is 8”. To be symbolic, it is written as;

\(\lim \limits_{x \to 2} (4x) = 4 \times 2 = 8\)

Continuity is another widespread topic in calculus. The easy method to test for the continuity of a function is to examine whether the graph of a function can be traced by a pen without lifting the pen from the paper. When you are doing with precalculus and calculus, a conceptual definition is almost sufficient but for higher level, a technical definition is required. You can learn a better and precise way of defining continuity by using limits.

Also, read:

Continuity Definition

A function is said to be continuous at a particular point if the following three conditions are satisfied.

  1. f(a) is defined
  2. \(\lim\limits_{x \to a} f(x)\) exists
  3. \(\lim\limits_{x \to a^{+}}f(x)= \lim\limits_{x \to a^{-}}f(x)= f(a)\)

A function is said to be continuous if you can trace its graph without lifting the pen from the paper. But a function is said to be discontinuous when it has any gap in between. Let us see the types discontinuities.

Types of Discontinuity

There are basically two types of discontinuity:

  • Infinite Discontinuity
  • Jump Discontinuity

Infinite Discontinuity

A branch of discontinuity wherein, a vertical asymptote is present at x = a and f(a) is not defined. This is also called as Asymptotic Discontinuities. If a function has values on both sides of an asymptote, then it cannot be connected, so it is discontinuous at the asymptote.

Jump Discontinuity

A branch of discontinuity wherein \(\lim\limits_{x \to a^{+}}f(x) \neq \lim\limits_{x \to a^{-}}f(x)\), but both the limits are finite. This is also called simple discontinuity or continuities of first kind.

Positive Discontinuity

A branch of discontinuity wherein a function has a pre-defined two-sided limit at x=a, but either f(x) is undefined at a or its value is not equal to the limit at a.

Limit Definition

A limit of a function is a number that a function reaches as the independent variable of the function reaches a given value. The value (say a) to which the function f(x) gets close arbitrarily as the value of the independent variable x becomes close arbitrarily to a given value a symbolized as f(x) = A.

Points to remember:

  • If limx→a- f(x) is the expected value of f at x = a given the values of ‘f’ near x to the left of a. This value is known as left-hand limit of ‘f’ at a.
  • If limx→a+ f(x) is the expected value of f at x = a given the values of ‘f’ near x to the right of a. This value is known as the right-hand limit of f(x) at a.
  • If the right-hand and left-hand limits coincide, we say the common value as the limit of f(x) at x = a and denote it by limx→a f(x).

One-Sided Limit

The limit that is based completely on the values of a function taken at x -value that is slightly greater or less than a particular value. A two-sided limit \(\lim\limits_{x \to a}f(x)\) takes the values of x into account that are both larger than and smaller than a. A one-sided limit from the left \(\lim\limits_{x \to a^{-}}f(x)\) or from the right \(\lim\limits_{x \to a^{-}}f(x)\) takes only values of x smaller or greater than a respectively.

Properties of Limit

  • The limit of a function is represented as f(x) reaches L as x tends to limit a, such that; limx→af(x) = L
  • The limit of the sum of two functions is equal to the sum of their limits, such that: limx→a [f(x) + g(x)] =  limx→a f(x) + limx→a g(x)
  • The limit of any constant function is a constant term, such that, limx→a C = C
  • The limit of product of the constant and function is equal to the product of constant and the limit of the function, such that: limx→a m f(x) = m limx→a f(x)
  • Quotient Rule: limx→a[f(x)/g(x)] =  limx→af(x)/limx→ag(x); if limx→ag(x) ≠ 0

Examples

1). To Compute \(\lim\limits_{x \to -2} \left ( 3x^{2}+5x-9 \right )\)

Solution:

First, use property 2 to divide the limit into three separate limits. Then use property 1 to bring the constants out of the first two. This gives,

\(\lim\limits_{x \to -2} \left ( 3x^{2}+5x-9 \right ) = \lim\limits_{x \to -2}(3x^{2}) + \lim\limits_{x \to -2}(5x) -\lim\limits_{x \to -2}(9)\)

\(= 3(-2)^{2} + 5(-2) -(9)\)

= -7

2). Find the limits of limx→3 [x(x+2)].

Solution: limx→3 [x(x+2)] = 3(3+2) = 3 x 5 = 15

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