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

Read in the blogs more about the unique learning experience at nubtrek.continue

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Limit of Functions with graph of numerator and denominator

» Slopes at the point `x=a` decide the limits of `f(x)` at `x=a`.

→ slope of numerator at `x=0` is `1`

→ slope of denominator at `x=0` is `1`

→ Both LHL and RHL limits `=1/1 = 1`

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

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

*simple steps to build the foundation*

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*simple steps to build the foundation*

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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.

Starting on learning "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.

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`. 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 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`. 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)`

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`. 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)`

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`. What parameter of the line defines the values of denominator at `1-delta` and `1+delta`?

- 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`. 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`
- 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))`.

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**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.

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

*your progress details*

Progress

*About you*

Progress

The limit of a function is computed when the function evaluates to indeterminate value 0 by 0 at x=a. Seeing the division in indeterminate value 0 by 0 , the function can be given as f of x = f n of x divided by f d of x, such that f n of x at x = a equals 0, and f d of x at x = a equals 0. ;; f n of x is the numerator;; f d of x is the denominator;;

Consider the function f of x = x squared minus 1 by x minus 1. The numerator and denominator evaluate to 0 at x = 1. Could you identify which graph represents the numerator and which graph represents the denominator?

1

2

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 of x = x squared minus 1 by x minus 1. The vertical purple lines, show 1 minus delta and 1+delta. What will be the value of denominator at 1 minus delta and 1+delta?

1

minus delta and plus delta

2

plus delta and minus delta

The answer is "minus delta and plus delta"

Given the graphs of numerator and denominator of the function f of x = x squared minus 1 by x minus 1. The vertical purple lines, show 1 minus delta and 1+delta. What will be the value of numerator at 1 minus delta and 1+delta?

1

plus 2 delta + delta squared and minus 2 delta + delta squared

2

minus 2 delta + delta squared and plus 2 delta + delta squared

The answer is "minus 2 delta + delta squared and plus 2 delta + delta squared"

Given the graphs of numerator and denominator of the function f of x = x squared minus 1 by x minus 1. The vertical purple lines, show 1 minus delta and 1+delta. ;; What parameter of the line defines the values of denominator at 1 minus delta and 1+delta?

slope

slope of the line

position

position of the line

The answer is ""

Given the graphs of numerator and denominator of the function f of x = x squared minus 1 by x minus 1. The vertical purple lines, show 1 minus delta and 1+delta. ;; Considering the curve at position x=1 as piecewise linear, What parameter of the curve defines the values of numerator at 1 minus delta and 1+delta

slope

slope of the curve at that position x=1

position

position of the line in the graph

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

Given the function f of x = f n of x by f d of x such that, f n of x at x = a, = 0, and f_d of x at x = a, = 0. ;; f of x at x=a+delta ;; equals f n of x at x=a+delta divided by f d of x at x=a+delta ;; equals f n of x at x=a+delta, minus f n of a, divided by, f d of x at x=a+delta, minus f d of a ;; as f n of a and f d of a are 0 ;; equals f n of x at x=a+delta, minus f n of a, by delta, divided by, f d of x at x=a+delta, minus f d of a, by delta ;; equals slope of f n of x at x=a, divided by, slope of f d of x at x=a;

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

Geometrical representation of Limits: If f of x = f n of x, by, f d of x, where f of x at x = a, equals 0 by 0;; f n of x at x = a, equals 0, and ;; f d of x at x = a, equals 0, then ;; the slopes on the left of x=a define the left hand limit ;; the slopes on the right of x=a define the right hand limit ;; The slopes referred are for the numerator and denominator.