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Thought-Process to Discover Knowledge

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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.
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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