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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 piecewise functions

With an example, calculation of limits for a piecewise function is discussed. The conditions under which a function is defined are illustrated with examples.



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Given piecewise function
`f(x)={(1, quad text(for)quad x>=1),(0.5, quad text(for)quad x<1):}`
What is `f(x)|_(x=1)`?limit of not differentiable function

  • `0.5`
  • `-1`
  • `+1`
  • `+1`

The answer is '`1`', as given by definition of the function.

Given function
`f(x)={(1, quad text(for)quad x>=1),(0.5, quad text(for)quad x<1):}`
what is left-hand-limit `lim_(x->1-)f(x)`?left-hand-limit of not differentiable function

  • `0.5`
  • `0.5`
  • `-1`
  • `+1`

The answer is '`0.5`'

`lim_(x->1-)f(x)`
`quad quad = f(x)|_(x=1-delta)`
`quad quad = 0.5 quad quad quad quad` (as `1-delta<1`)

Given function
`f(x)={(1, quad text(for)quad x>=1),(0.5, quad text(for)quad x<1):}`
what is right-hand-limit `lim_(x->1+)f(x)`?right-hand-limit of not differentiable function

  • `0.5`
  • `-1`
  • `+1`
  • `+1`

The answer is '`1`'

`lim_(x->1+)f(x)`
`quad quad = f(x)|_(x=1+delta)`
`quad quad = 1 quad quad quad quad` (as `1+delta>1`)

Given function
`f(x)={(1, quad text(for)quad x>=1),(0.5, quad text(for)quad x<1):}`

 •  `f(1)=1`

 •  `lim_(x->1-)f(x) = 0.5`

 •  `lim_(x->1+)f(x) = 1`
not differentiable function limits The function is not continuous at `x=1`. But, the function is defined by the given definition of the function.

If the function evaluates to a real number, then the function is defined at that input value. The limit may not be defined.

Function is defined by value: If `f(a) in RR`, and `lim_(x->a-)f(x)` `!= lim_(x->a+)f(x)`, then function is not continuous but defined by the definition of the function.

Solved Exercise Problem:

Given function
`f(x)={(2x, quad text(for)quad x>2),(x^2, quad text(for)quad x=2),(|x|+2, quad text(for)quad x<2):}`
Examine the function at `x=2`.

  • continuous
  • defined by limit
  • defined by value
  • All the above
  • All the above

The answer is 'All the above'.

                            
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