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



Limits by Numerator and Denominator

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

 »  eg: `f(x) = color(deepskyblue)(2x-4)/color(coral)(x-2)`
    →  slope of numerator at `x=2` is `2`

    →  slope of denominator at `x=2` is `1`

    →  Both LHL and RHL limits `=2/1 = 2`

Understanding Limits : Examples

plain and simple summary

nub

plain and simple summary

nub

dummy

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.




More examples are explained to evaluate limit of a function, using the graphs of the numerator and denominator.


Keep tapping on the content to continue learning.
Starting on learning "examples to understand limits" ;; More examples are explained to evaluate limit of a function, using the graphs of the numerator and denominator.

Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)`. limit of linear slopes What will be the value of `lim_(x->2) f(x)`?

  • `=1`
  • `>1`
  • `<1`

The answer is '`>1`'. The slope of numerator is greater than the denominator.

Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)`.limit of linear slopes The function given is `f(x) = color(deepskyblue)(2x-4)/color(coral)(x-2)`. The slope of the numerator is `2` and slope of the denominator is `1`.

Left-hand-limit `x=2-delta` of
`color(deepskyblue)(2x-4)/color(coral)(x-2)`
`quad quad = color(deepskyblue)(2(2-delta)-4)/color(coral)(2-delta-2)`
`quad quad = color(deepskyblue)(4-2delta-4)/color(coral)(-delta)`
`quad quad = color(deepskyblue)(-2delta)/color(coral)(-delta)`
`quad quad = 2`

Similarly, the right-hand-limit can be worked out to `2`.

Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)`. limit of negative slopes What will be the value of `lim_(x->2) f(x)`?

  • `=1`
  • `<-1`
  • `>1`

The answer is '`<-1`'. The slope of the numerator is negative and decreasing steeper than the denominator.

Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)`.limit of negative slopes The function given is `f(x) = color(deepskyblue)(6-3x)/color(coral)(.5x-1)`. The slope of the numerator is `-3` and the slope of the denominator is `0.5`.

Right-hand-limit `x=2+delta`
`color(deepskyblue)(6-3(2+delta))/color(coral)(0.5(2+delta)-1)`
`quad quad = color(deepskyblue)(-3delta)/color(coral)(0.5delta)`
`quad quad = -6`

Similarly, left hand limit can be worked out as `-6`.

Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)`. limit of sin(x)/x slopes What will be the `lim_(x->0) f(x)`?

  • `=1`
  • `>1`
  • `<1`

The answer is '`=1`'. The slope of numerator equals that of the denominator.

Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)` (in purple).limit of sin(x)/x slopes The function given is `color(purple)(f(x)) = color(deepskyblue)(sin x)/color(coral)(x)`. At `x=0` the slope of the numerator is `1` and slope of the denominator is `1`.

Proof for limit of this function is explained later.

Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)`. limit of (1-cos(x))/x slopes What will be the `lim_(x->0) f(x)`?

  • `oo`
  • `>1`
  • `<1`
  • `=0`

The answer is '`=0`'. The slope of numerator is `0` and that of the denominator is `1`.

Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)` (in purple).limit of (1-cos(x))/x slopes The function given is `f(x) = color(deepskyblue)(1- cos x)/color(coral)(x)`. The slope of the numerator is `0` and the slope of the denominator is `1`.

Proof for limit of this function is explained later.

Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)`. limit of 1/x slopes What will be the `lim_(x->0+) f(x)`?

  • `=oo`
  • `=1`
  • `<1`
  • `=0`

The answer is '`=oo`'. This function is not a candidate to analyze numerator and denominator, as the function does not evaluate to `0/0` form.

Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)` (in purple).limit of linear slopes The function given is `f(x) = color(deepskyblue)(1)/color(coral)(x)`.
This function does not satisfy the pre-condition that numerator and denominator has to evaluate to `0`. So the analysis by comparing slopes is not applicable.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy


           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)`. limit of (1-cos(x))/x slopes What will be the `lim_(x->0) f(x)`?

  • `oo`
  • `>1`
  • `<1`
  • `=0`

The answer is '`0`'. The slope of numerator is `0` and that of the denominator is `1`.

your progress details

Progress

About you

Progress

Consider the graphs of numerator in blue and denominator in orange of a function f of x. What will be the value of limit x tending to 2, f of x?
equal;equals
equals 1
greater
greater than 1
less
less than 1
The answer is "greater than 1". The slope of numerator is greater than the denominator.
Consider the graphs of numerator in blue and denominator in orange of a function f of x. ;; The function given is f of x = 2x minus 4 by x minus 2. The slope of the numerator is 2 and slope of the denominator is 1. ;; left hand limit x = 2 minus delta of 2 x minus 4 by x minus 2 ;; equals 2 into 2 minus delta minus 4, by , 2 minus delta minus 2;; equals 4 minus 2 delta minus 4, by, minus delta ;; equals minus 2 delta by minus delta;; equals 2;; Similarly, the right-hand-limit can be worked out to 2.
Consider the graphs of numerator in blue and denominator in orange of a function f of x. What will be the value of limit x tending to 2, f of x?
equals;equal
equals 1
less;minus
less than minus 1
greater
greater than 1
The answer is "less than minus 1". The slope of the numerator is negative and decreasing steeper than the denominator.
Consider the graphs of numerator in blue and denominator in orange of a function f of x.;; The function given is f of x = 6 minus 3 x divided by point 5 x minus 1. The slope of numerator is minus 3 and the slope of the denominator is point 5. Right hand limit is worked out as minus 6. Similarly, left hand limit can be worked out as minus 6.
Consider the graphs of numerator in blue and denominator in orange of a function f of x. What will be the value of limit x tending to 0, f of x?
equal;equals
equals 1
greater
greater than 1
less
less than 1
The answer is "equals 1". The slope of the numerator equals that of the denominator.
Consider the graphs of numerator in blue and denominator in orange of a function f of x in purple.;; The function given is f of x = sine x by x. ;; At x = 0; the slope of the numerator is 1 and the slope of the denominator is 1. ;; Proof for limit of this function is explained later.
Consider the graphs of numerator in blue and denominator in orange of a function f of x. What will be the value of limit x tending to 0, f of x?
infinity
infinity
greater
greater than 1
less
less than 1
equals;equal;0
equals 0
The answer is "equals 0". The slope of the numerator is 0 and that of the denominator is 1.
Consider the graphs of numerator in blue and denominator in orange of a function f of x in purple.;; the function given is f of x = 1 minus cos x divided by x. The slope of the numerator is 0 and the slope of the denominator is 1. ;; Proof for limit of this function is explained later.
Consider the graphs of numerator in blue and denominator in orange of a function f of x. What will be the value of limit x tending to 0+, f of x?
infinity
equals infinity
1
equals 1
less
less than 1
0
equals 0
The answer is "equals infinity. ". The function is not a candidate to analyze numerator and denominator, as the function does not evaluate to 0 by 0 form.
Consider the graphs of numerator in blue and denominator in orange of a function f of x in purple. ;; the function given is f of x = 1 by x. ;; This function does not satusfy the pre-condition that numerator and denominator has to evaluate to 0. So the analysis by comparing slopes is not applicable.
Consider the graphs of numerator in blue and denominator in orange of a function f of x. What will be the value of limit x tending to 0, f of x?
infinity
infinity
greater
greater than 1
less;1
less than 1
equals; equal;0
equals 0
The answer is "equals 0"

we are not perfect yet...

Help us improve