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.

Continuity:-

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.

Limit:

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.

Example:

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

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.

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Practise This Question

Calculate 32f(x)dx where
f(x)={6ifx>13x2ifx1