evaluating a function

Consider the value of function `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` when `x=2.2`.

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

function evaluating to 0/0

Consider the value of function `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` when `x=1`

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)`

That is, the function evaluates to indeterminate value when `x=1`.

Let us closely examine the function `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)`.

The numerator be factorized as `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)`.

It is noted that the function can be 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`.

another example

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

The answer is '`0/0`'

Outline

The outline of material to learn "limits (calculus)" is as follows.

Note : click here for detailed outline of Limits(Calculus).

    →   Indeterminate and Undefined

    →   Indeterminate value in Functions

    →   Expected Value

    →   Continuity

    →   Definition by Limits

    →   Geometrical Explanation for Limits

    →   Limit with Numerator and Denominator

    →   Limits of Ratios - Examples

    →   L'hospital Rule

    →   Examining a function

    →   Algebra of Limits

    →   Limit of a Polynomial

    →   Limit of Ratio of Zeros

    →   Limit of ratio of infinities

    →   limit of Binomial

    →   Limit of Non-algebraic Functions
next