Limits, Continuity and Differentiability: What Are Limits: Existence of Limit (For CBSE, ICSE, IAS, NET, NRA 2022)

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Title: Limits, Continuity and Differentiability

  • The limit concept is certainly indispensable for the development of analysis, for convergence and divergence of infinite series also depends on this concept.
  • Theory of limits and then defining continuity, differentiability and the definite integral in terms of the limit concept is successfully executed by mathematicians.

What Are Limits?

  • Limit of a function may be a finite or an infinite number.
  • If it just implies that the function f (x) tends to assume extremely large positive values in the vicinity of i.e..

  • A function is said to be indeterminate at any point if it acquires one of the following values at that particular point
  • The form is the standard indeterminate form.
  • The point ‘∞’ cannot be plotted on the paper. It is just a symbol and not a number.
  • Infinity (∞) does not obey the laws of elementary algebra.

Existence of Limit

The limit will exist if the following conditions get fulfilled:

Both LHS and RHS should be finite

Some Important Limits

Continuity

  • A continuous function is a function for which small changes in the input results in small changes in the output. Otherwise, a function is said to be discontinuous.
  • A function f (x) is said to be continuous at x = a if
  • i.e.. L. H. L = R. H. L = value of the function at x = a
  • A function f (x) is said to be discontinuous Function.

Differentiability

Existence of Derivative

  • Right and left hand derivative
  • A piecewise function is differentiable at a point if both of the pieces have derivatives at that point, and the derivatives are equal at that point.
  • In this case, Sal took the derivatives of each piece: first he took the derivative of at x = 3 and saw that the derivative there is 6.
  • A function f is said to be continuously differentiable if the derivative f′ (x) exists and is itself a continuous function.
  • Though the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity.
  • For example, the function

  • is differentiable at 0,

Differentiable Function

Differentiable Function
  • A differentiable function of one real variable is a function whose derivative exists at each point in its domain.
  • The graph of a differentiable function has a non-vertical tangent line at each interior point in its domain.
  • A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp.

How Can a Function Fail to be Differentiable?

The function f (x) is said to be non-differentiable at x = a if

  • Both R. H. D & L. H. D exist but not equal
  • Either or both R. H. D & L. H. D are not finite
  • Either or both R. H. D & L. H. D do not exist.

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