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Thought-Process to Discover Knowledge

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The right-hand and left-hand limits are equal for most input values for most functions. This is commonly referred as limit of a function.

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Given that function `f(x)` evaluates to indeterminate value at `x=a`. To evaluate the expected value of `f(x)|_(x=a)`, we examine ;

 •  Left-hand-limit `lim_(x->a-) f(x)`

 •  Right-hand-limit `lim_(x->a+) f(x)`

If these two limits are equal then the result is referred as "limit of the function at the input value" `lim_(x->a) f(x)`

The significance of this is that, most functions have both right-hand-limit and left-hand-limit equal.

When left-hand-limit and right-hand-limit are equal, the limits are referred together as limit of a function.

Limit of a function: Given function `f(x)` and that `f(x)|_(x=a) = 0/0`.
If `lim_(x->a+) f(x) = lim_(x->a-) f(x)`,
then the common value is referred as limit of the function `lim_(x->a) f(x)`.

Solved Exercise Problem:

If a function `f(x)` is discontinuous at `x=a`, then what is `lim_(x->a) f(x)`?

  • left-hand-limit
  • right-hand-limit
  • `f(a)`
  • cannot be computed
  • cannot be computed

The answer is 'cannot be computed'. It is given that the function is discontinuous at `x=a`, and that implies left-hand-limit and right-hand-limits are not equal. In that case, limit of the function cannot be computed without specifying left or right.

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