Limits And Continuity

Limits are one of the highlighted concepts to understand if you are preparing calculus. 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

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

In this article, we will define a limit along with examples on a limit of functions.

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 a continuity by using limits.


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.

Types of Continuity:

Infinite Discontinuity:

A branch of discontinuity wherein, a vertical asymptote is present at x = a and f(a) is not defined.

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.


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.

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.


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


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

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.

To know more about Limits and Continuity, Calculus, Differentiation etc. and solved examples, visit our sit BYJU’S.

Practise This Question

The prime factorization of a number is given as 3×3×11×101. Which of the following is correct about the number.

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