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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue
mathsLimit of a functionBasics: Limit of a function

Function defined at an input value

The two conditions, to figure out if a function is defined at an input value or not, are explained.



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Under which of the following conditions, a function `f(x)` is defined at the input value `x=a`?

  • when `f(a)` evaluates to a defined value
  • when `f(a)` evaluates to a defined value
  • when `f(a)` is `oo`
  • when `f(a)` is `0/0`
  • all the above

The answer is 'when `f(a)` evaluates to a defined value'.

If `f(a)` evaluates to `oo` or `0/0`, then the function need to be further examined.

If the left-hand-limit and right-hand-limit take a defined value and are equal, then the function is defined at that input value.

This is understood from

 •  In an application scenario, the value is not exact and so the function will evaluate to a stable value given by limits.

 •  In an abstract scenario, the expected value at that input value can be taken to be the value of the function.

A function is defined at an input value, if the function evaluates to a definite value OR if the left-hand-limit and right-hand-limits are equal.

What does the word 'defined' mean?

  • give exactly; state or describe without any ambiguity
  • give exactly; state or describe without any ambiguity
  • extremely loud; blaring in high pitch

The answer is 'give exactly; state or describe without any ambiguity'.

Function is defined: Given a function `f(x)`, the function is defined at `x=a`
 •  if `f(a)` is a defined value. OR

 •  if `f(a)` is an indeterminate value `0/0` or undefined large `oo`, then `lim_(x->a+)f(x)=lim_(x->a-)f(x)`

Difference between Defined and Continuous :
A function is defined if it evaluates to a definite value OR when is the function evaluates to indeterminate value `0/0`, then left-hand-limit and right-hand-limit are equal.

A function is continuous only when it evaluates to a definite value at the input value AND also left-hand-limit and right-hand-limit are equal to the definite value at the input value.

 •  For a function to be continuous at a input value, it has to be defined at that input value.

 •  If a function is defined at an input value, it does not imply it is continuous. It may be or may not be continuous.

 •  If a function is not defined at an input value, it implies it is not continuous.

Solved Exercise Problem:

If a function is not continuous at an input value, can the function be defined at that input value?

  • Yes.
  • Yes.
  • No.

The answer is 'Yes.'. Theoretically a function can have a value `f(a)` different to the limits.

                            
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