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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

Limit of functions with abrupt change

With an example, calculation of limits for a function with special property is discussed.



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Given function `f(x)=|x-1|`. what is `f(x)|_(x=1)`?limit of not differentiable function

  • `0`
  • `0`
  • `-1`
  • `+1`

The answer is '`0`'

Given function `f(x)=|x-1|`. what is left-hand-limit `lim_(x->1-)f(x)`?left-hand-limit of not differentiable function

  • `0`
  • `0`
  • `-1`
  • `+1`

The answer is '`0`'

The left-hand-limit `lim_(x->1-)f(x)`
`quad quad = |1-delta-1|`
`quad quad = |-delta|`
`quad quad = delta`
`quad quad = 0 quad quad quad quad` substituting `delta=0`

Given function `f(x)=|x-1|`. what is right-hand-limit `lim_(x->1+)f(x)`?right-hand-limit of not differentiable function

  • `0`
  • `0`
  • `-1`
  • `+1`

The answer is '`0`'.

The right-hand-limit `lim_(x->1+)f(x)`
`quad quad = |1+delta-1|`
`quad quad = |delta|`
`quad quad = delta`
`quad quad = 0 quad quad quad quad` substituting `delta=0`

Given function `f(x)=|x-1|`.

 •  `f(1)=0`

 •  `lim_(x->1-)f(x) = 0`

 •  `lim_(x->1+)f(x) = 0`
not differentiable function limits The function is continuous at `x=1`. Note that the function undergoes an abrupt change at `x=1`.

 •  for `x<1`, the slope of the line is `-1`

 •  for `x>1`, the slope of the line is `1`

 •  At `x=1`, the slope of the line is not defined.

Solved Exercise Problem:

Given the function `f(x)=|sin x|`. Is the function continuous or defined at `x=pi`?

  • continuous
  • defined and not continuous
  • defined and not continuous
  • not defined and not continuous

The answer is 'continuous'

                            
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