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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 functionCalculating Limits

Examining Function at an input value

Revising the basics learned in earlier lesson and summarizing the understanding.



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A function `f(x)` is examined at an input value `x=a`. Which of the following is to be examined?

  • `f(a)`
  • left-hand-limit
  • right-hand-limit
  • all the above
  • all the above

The answer is 'all the above'.

It is important to note that the L'Hospital's rule is applicable only if the limit exists. The discussion in this topic is about finding if the limit exists or not. So the L'Hospital's rule is not used in finding the limits.

When a function is examined at an input value, what are the possible deductions one can make about the function at that input value?

  • Function is continuous
  • Function is not continuous but defined
  • Function is not continuous and not defined
  • Any one of the above
  • Any one of the above

The answer is 'Any one of the above'.

Whether a function is continuous or defined or not defined is determined by examining
 • value at the input value,

 • left-hand-limit at the input value, and

 • right-hand-limit at the input value.

Describing a function: A function `f(x)` is evaluated for three values to describe at an input value `x=a`

 •  `f(a)`

 •  `lim_(x->a-) f(x)`

 •  `lim_(x->a+) f(x)`

Solved Exercise Problem:

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

  • Yes. If the limits are equal.
  • No. Irrespective of limits if the function is not defined at an input value, it is not continuous.
  • No. Irrespective of limits if the function is not defined at an input value, it is not continuous.

The answer is 'No. Irrespective of limits if the function is not defined at an input value, it is not continuous at that input value.'

                            
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