nubtrek

Server Error

Server Not Reachable.

This may be due to your internet connection or the nubtrek server is offline.

Thought-Process to Discover Knowledge

Welcome to nubtrek.

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.

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

User Guide   

Welcome to nubtrek.

The content is presented in small-focused learning units to enable you to
  think,
  figure-out, &
  learn.

Just keep tapping (or clicking) on the content to continue in the trail and learn. continue

User Guide   

To make best use of nubtrek, understand what is available.

nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

  nub,

  trek,

  jogger,

  exercise.

continue

User Guide    

nub is the simple explanation of the concept.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about. continue

User Guide    

trek is the step by step exploration of the concept.

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step. continue

User Guide    

jogger provides the complete mathematical definition of the concepts.

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen. continue

User Guide    

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge. continue

summary of this topic

Understanding limits with Graphs

Understanding limits with Graphs

Voice  

Voice  



Home



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`

Understanding limits with Graphs of Numerator and Denominator

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

trek

simple steps to build the foundation

trek

Support Nubtrek     
 

You are learning the free content, however do shake hands with a coffee to show appreciation.
To stop this message from appearing, please choose an option and make a payment.




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.


Keep tapping on the content to continue learning.
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`.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 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)`

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)`

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
  • 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`
  • 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.

we are not perfect yet...

Help us improve