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Basics: Limit of a function

Basics: Limit of a function

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Indeterminate Value in Functions


 »  Functions evaluate to `0/0`
    →  eg: `(x^2-4)/(x-2)|_(x=2) = 0/0`

Though the function evaluates to `0/0`, it may evaluate to a value.

Indeterminate value in functions

plain and simple summary

nub

plain and simple summary

nub

dummy

Functions can evaluate to indeterminate value `0/0` at some input values.

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simple steps to build the foundation

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


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Starting on learning "indeterminate value in functions". ;; Some functions evaluate to indeterminate value at some input values. This is illustrated with an example.

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`

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

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)`
  • 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`

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`.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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

  • `0/0`
  • `10`
  • `1`

The answer is '`0/0`'

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What is the value of f of x equals x squared minus 1 by x minus 1 when x equals 2 point 2?
0
0 by 0
indeterminate;value
indeterminate value
3;point;2;3.2
3 point 2
The answer is "3 point 2". On substituting x equals 2. point 2, we get f of 2 point 2 equals 2 point 2 squared minus 1 by 2 point 2 minus 1.
What is the value of f of x equals x squared minus 1 by x minus 1 when x equals 1?
0
0 by 0
indeterminate;value
indeterminate value
both;above
both the above
The answer is "Both the Above". On substituting x=1, we get ;; f of 1 ; equals 1 squared minus 1 by 1 minus 1 ;; equals 0 by 0.
Let us closely examine the function f of x equals x squared minus 1 by x minus 1. Can the numerator be factorized?
yes;s;squared;minus;1
yes; x squared minus 1 equals x plus 1 multiplied x minus 1
cannot;be;factorized;not;only
Cannot be factorized as Factorization is only for numbers, not for algebraic expressions
The answer is "Yes; x squared minus 1 equals x plus 1 multiplied x minus 1"
Rewriting the function f of x equals x squared minus 1 by x minus 1 as the function ;; f of x equals x + 1 multiplied x minus 1 divided by x minus 1. ;; Can the function be simplified?
simplified to; simplified 2
Simplified to f of x equals x + 1
only;when;not;equal;1
simplified only when x not equal 1 to f of x equals x + 1
The answer is "simplified only when x not equal 1 to f of x equals x + 1". Note that 0 cannot be canceled out in expressions or equations.
So the given function ;; f of x ;; equals x + 1 when x not equals 1 ;; and x squared minus 1 by x minus 1 when x equals 1 ;; By this it is concluded that f of x at x =1 is indeterminate value 0 by 0.;; Many students wrongly understand that the algebraic simplified, like canceling x minus 1 in the example above, solves the indeterminate value. It is not so, The function remains indeterminate at that input value x equals 1.
Functions can evaluate to indeterminate value 0/0 at some input values.
Function evaluates to indeterminate value: Function f of x evaluates to indeterminate value for x equals a if f of a equals 0 by 0.
Given f of x equal x cube minus 8 by x squared minus 4. What is f of 2?
0
0 by 0
10
10
1
1
The answer is "0 by 0"

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