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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.

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mathsLimit of a functionBasics: Limit of a function

### Limit of a function

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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