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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 functionBasics: Limit of a function

### Indeterminate value in functions

Some functions evaluate to indeterminate value at some input values. This is illustrated with an example.

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What is the value of f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1) when x=2.2?

• 0/0
• indeterminate value
• 3.2
• 3.2

The answer is '3.2'. On substituting x=2.2, we get f(2.2)= color(deepskyblue)(2.2^2-1)/color(coral)(2.2-1).

What is the value of f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1) when x=1?

• 0/0
• indeterminate value
• both the above
• both the above

The answer is 'Both the above'. On substituting x=1, we get
f(1)
quad quad =color(deepskyblue)(1^2-1)/color(coral)(1-1)
quad quad = color(deepskyblue)(0)/color(coral)(0).

Let us closely examine the function f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1). Can the numerator be factorized?

• Yes, x^2-1 = (x+1)(x-1)
• Yes, x^2-1 = (x+1)(x-1)
• Cannot be factorized as Factorization is only for numbers, not for algebraic expressions

The answer is 'Yes - x^2-1 = (x+1)(x-1)'.

Rewriting the function f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1) as the function f(x) = color(deepskyblue)((x+1)(x-1))/color(coral)(x-1). Can the function be simplified?

• Simplified to f(x)= x+1
• Simplified only when x!=1 to f(x)= x+1
• Simplified only when x!=1 to f(x)= x+1

The answer is 'Simplified to f(x)= x+1 when x!=1'. Note that 0 cannot be canceled out in expressions or equations.

So the given function
f(x)
quad quad = x+1  when x!=1
quad quad = (x^2-1)/(x-1)  when x=1

By this it is concluded that f(x)|_(x=1) is indeterminate value 0/0.

Many students wrongly understand that the algebraic simplification (like canceling x-1 in the example above) solves the indeterminate value. It is not so -- the function remains indeterminate at that input value x=1.

Function evaluates to indeterminate value: Function f(x) evaluates to indeterminate value for x=a if f(a) = 0/0.

Solved Exercise Problem:

Given f(x)=(x^3-8)/(x^2-4) What is f(2)?

• 0/0
• 0/0
• 10
• 1

The answer is '0/0'

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