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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.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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mathsLimit of a functionCalculating Limits

### Limit of defined functions

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

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Given function f(x)=color(deepskyblue)(x^2-1)/color(coral)(x-1). what is f(x)|_(x=1)?

• Substitute x=1
• (1-1)/(1-1)
• 0/0
• all the above
• all the above

The answer is 'All the above'. The options provide the steps to evaluate f(1).

Given function f(x)=color(deepskyblue)(x^2-1)/color(coral)(x-1). what is left-hand-limit lim_(x->1-)f(x)?

• 2
• 2
• 0/0
• -delta
• -2

The answer is '2'.

lim_(x->1-)f(x)
quad quad = color(deepskyblue)((1-delta)^2-1)/color(coral)((1-delta)-1)
quad quad = color(deepskyblue)(1-2delta+delta^2-1)/color(coral)(1-delta-1)
quad quad = color(deepskyblue)(-delta(2-delta)) /color(coral)(-delta)
quad quad = 2-delta
quad quad = 2 (substituting delta=0)

Given function f(x)=color(deepskyblue)(x^2-1)/color(coral)(x-1). what is right-hand-limit lim_(x->1+)f(x)?

• 2
• 0/0
• 0/0
• delta
• -2

The answer is '2'.

lim_(x->1+)f(x)
quad quad = color(deepskyblue)((1+delta)^2-1)/color(coral)((1+delta)-1)
quad quad = color(deepskyblue)(1+2delta+delta^2-1)/color(coral)(1+delta-1)
quad quad = color(deepskyblue)(delta(2+delta)) /color(coral)(delta)
quad quad = 2+delta
quad quad = 2 (substituting delta=0)

Given function f(x)=color(deepskyblue)(x^2-1)/color(coral)(x-1).

•  f(1)=0/0 indeterminate value

•  lim_(x->1-)f(x) = 2

•  lim_(x->1+)f(x) = 2
The function is defined at x=1.

When left-hand-limit and right-hand-limit are equal, then limit is defined.
Limit is referred as "limit of the function"

If the left hand and right hand limits are equal then the limit is defined at that input value.

If the function evaluates to indeterminate value and the limit is defined, then the function is defined by the limits.

Limit is defined: If lim_(x->a-)f(x) = lim_(x->a+)f(x) then the limits are commonly given as lim_(x->a) f(x)
Function is defined by Limit: If f(a) !in RR and lim_(x->a-)f(x) = lim_(x->a+)f(x), then the function is defined by limit at x=a.

Solved Exercise Problem:

Find if the function f(x)=color(deepskyblue)(3x^2-2x-8)/color(coral)(x-2) is continuous or defined at x=2?

• Both continuous and defined
• Not defined
• Not Continuous and Defined
• Not Continuous and Defined

The answer is 'Not Continuous and Defined'

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