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Thought-Process to Discover Knowledge

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mathsLimit of a functionUnderstanding limits with Graphs

Understanding limits with Graphs of Numerator and Denominator

The function for which limit is computed is considered as two constituent functions of numerator and denominator. The limit of the function is explained with the graphs of numerator and denominator.



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The limit of a function is computed when the function evaluates to indeterminate value `0/0` at `x=a`. Seeing the division in indeterminate value `0/0`, the function can be given as `f(x)=color(deepskyblue)(f_n(x))/color(coral)(f_d(x))` such that `color(deepskyblue)(f_n(x))|_(x=a) = 0` and `color(coral)(f_d(x))|_(x=a) = 0`.

Consider the function `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)`. The numerator and denominator evaluate to `0` at `x=1`.plot of numerator and denominator Could you identify which graph represents the numerator and which graph represents the denominator?

  • numerator is the curve in blue, and denominator is the line in orange
  • numerator is the curve in blue, and denominator is the line in orange
  • numerator is the line in orange, and denominator is the curve in blue

The answer is ' numerator is the curve in blue, and denominator is the line in orange'

Given the graphs of numerator and denominator of the function `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)`. The vertical purple lines, show `1-delta` and `1+delta`. limit of numerator and denominator What will be the value of denominator at `1-delta` and `1+delta`?

  • `color(coral)(-delta)` and `color(coral)(+delta)`
  • `color(coral)(-delta)` and `color(coral)(+delta)`
  • `color(coral)(+delta)` and `color(coral)(-delta)`

The answer is '`-delta` and `+delta`'.

Given the graphs of numerator and denominator of the function `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)`. The vertical purple lines, show `1-delta` and `1+delta`. limit of numerator and denominator What will be the value of numerator at `1-delta` and `1+delta`?

  • `color(deepskyblue)(+2delta+delta^2)` and `color(deepskyblue)(-2delta+delta^2)`
  • `color(deepskyblue)(-2delta+delta^2)` and `color(deepskyblue)(+2delta+delta^2)`
  • `color(deepskyblue)(-2delta+delta^2)` and `color(deepskyblue)(+2delta+delta^2)`

The answer is '`-2delta+delta^2` and `+2delta+delta^2`'.

Given the graphs of numerator and denominator of the function `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)`. The vertical purple lines, show `1-delta` and `1+delta`. limit of numerator and denominator What parameter of the line defines the values of denominator at `1-delta` and `1+delta`?

  • slope of the line
  • slope of the line
  • position of the line

The answer is 'slope of the line'.

Given the graphs of numerator and denominator of the function `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)`. The vertical purple lines, show `1-delta` and `1+delta`. limit of numerator and denominator Considering the curve at position `x=1` as piecewise linear, What parameter of the curve defines the values of numerator at `1-delta` and `1+delta`?

  • slope of the curve at that position `x=1`
  • slope of the curve at that position `x=1`
  • position of the line in the graph

The answer is 'slope of the curve at that position'.

Given the function `f(x)=color(deepskyblue)(f_n(x))/color(coral)(f_d(x))` such that `color(deepskyblue)(f_n(x))|_(x=a) = 0` and `color(coral)(f_d(x))|_(x=a) = 0`.

`f(x)|_(x=a+delta)`
`quad quad = color(deepskyblue)(f_n(x))|_(x=a+delta) -: color(coral)(f_d(x))|_(x=a+delta)`
`quad quad = [color(deepskyblue)(f_n(x))|_(x=a+delta) - f_n(a)]`
`quad quad quad quad -: [color(coral)(f_d(x))|_(x=a+delta) - f_d(a)]`
as `f_n(a) = 0` and `f_d(a)=0`.

`quad quad = [color(deepskyblue)(f_n(x))|_(x=a+delta) - f_n(a)]/delta`
`quad quad quad quad -: [color(coral)(f_d(x))|_(x=a+delta) - f_d(a)]/delta`
`quad quad = color(deepskyblue)(text(slope) f_n(x)|_(x=a)) -: color(coral)(text(slope) f_d(x)|_(x=a))`.

Slope of numerator and denominator defines the limits of the function.

Geometrical representation of Limits: If `f(x)=color(deepskyblue)(f_n(x))/color(coral)(f_d(x))`, where
`f(x)|_(x=a) = 0/0`;
`color(deepskyblue)(f_n(x))|_(x=a) = 0` and
`color(coral)(f_d(x))|_(x=a) = 0`, then

 •  the slopes on the left of `x=a` define the left-hand-limit and

 •  the slopes on the right of `x=a` define the right-hand-limit.

The slopes referred are for the numerator and denominator.

                            
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