Server Not Reachable. *This may be due to your internet connection or the nubtrek server is offline.*

Thought-Process to Discover Knowledge

Welcome to **nub****trek**.

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

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

Welcome to **nub****trek**.

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

The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

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.

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

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

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

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

Voice

Voice

Home

Understanding Limits with Graph of the function

» Values of Function at `x=a`

→ Evaluated at input `f(x)|_(x=a)` or `f(a)`

→ Left-hand-limit `lim_(x->a-) f(x)`

→ Right-hand-limit `lim_(x->a+) f(x)`

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

Limit of a function at `x=a` is understood as the value of function at `x=a`, left side of that : `x=a-delta`, and right side of that : `x=a+delta`

*simple steps to build the foundation*

trek

*simple steps to build the foundation*

trek

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.*

Geometrical meaning of finding limit of a function is explained.

Starting on learning "Understanding limits with the graph of the function". ;; Geometrical meaning of finding limit of a function is explained.

In the previous pages, limit is defined in algebraic form.

In this topic, the function is considered as a graph in a 2D coordinate plane and the meaning of limit is explained.

What is the value of `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` when `x=1`?

- `0/0`
- indeterminate value
- both the above

The answer is 'Both the above'. On substituting `x=1`, we get `f(1)= 0/0`.

The plot of `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` is shown. At `x=1`, the graph breaks and the function does not evaluate to a real number.

Left-hand-limit of `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` is shown. At `x=1-delta`, dotted vertical line is shown.

Applying limit is moving the vertical line towards `x=1` and making `delta~=0`. This is shown as `lim_(x->1-)` in the figure.

`lim_(x->1-)f(x)`

`quad quad = color(deepskyblue)((1-delta)^2-1)/color(coral)((1-delta)-1)`

`quad quad = color(deepskyblue)(1-2delta+delta^2-1)/color(coral)(1-delta-1)`

`quad quad = color(deepskyblue)(-delta(2-delta)) /color(coral)(-delta)`

`quad quad = 2-delta`

`quad quad = 2` (substituting `delta=0`)

Right-hand-limit of `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` is shown. At `x=1+delta`, dotted vertical line is shown.

Applying limit is moving the vertical line towards `x=1` and making `delta~=0`. This is shown as `lim_(x->1+)` in the figure.

`lim_(x->1+)f(x)`

`quad quad = color(deepskyblue)((1+delta)^2-1)/color(coral)((1+delta)-1)`

`quad quad = color(deepskyblue)(1+2delta+delta^2-1)/color(coral)(1+delta-1)`

`quad quad = color(deepskyblue)(delta(2+delta)) /color(coral)(delta)`

`quad quad = 2+delta`

`quad quad = 2` (substituting `delta=0`)

Both the limits of `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` is shown. The right-hand-limit and left-hand-limits converge to `2`.

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

Limits of a function at `x=a` are illustrated in the figure.

• Evaluated at input `f(x)|_(x=a)` or `f(a)`

• Left-hand-limit `lim_(x->a-) f(x)`

• Right-hand-limit `lim_(x->a+) f(x)`

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

*your progress details*

Progress

*About you*

Progress

In the previous pages, limit is defined in algebraic form. ;; In this topic, the function is considered as a graph in a 2D coordinate plane and the meaning of limit is explained.

What is the value of f of x = x squared minus 1 by x minus 1 when x = 1?

by;0

0 by 0

indeterminate;value

indeterminate value

both;above

both the above

The answer is "Both the above". On substituting x = 1, we get f of 1 equals 0 by 0.

The plot of f of x = x squared minus 1 by x minus 1 is shown. At x=1, the graph breaks and the function does not evaluate to a real number.

Left-hand-limit of f of x = x squared minus 1 by x minus 1 is shown. ;; at x = 1 minus delta, dotted vertical line is shown ;; Applying limit is moving the vertical line towards x=1 and making delta very close to 0 or approximately 0. This is show as limit x tending to 1 minus in the figure;; limit x tending to 1 minus, f of x ;; equals 1 minus delta squared, minus 1 by, 1 minus delta minus 1;; equals 1 minus 2 delta + delta squared minus 1 by 1 minus delta minus 1 ;; equals minus delta into 2 minus delta by minus delta ;; equals 2 minus delta;; equals 2

Right hand limit of f of x = x squared minus 1 by x minus 1 is shown. ;; At x = 1 + delta, dotted vertical line is shown. ;; Applying limit is moving the vertical line towards x=1 and making delta approximately equals 0. This is shown as limit x tending to 1 plus in the figure. ;; limit x tending to 1 plus f of x;; equals 1 plus delta squared, minus 1 by 1 +delta minus 1 ;; equals 1+2 delta + delta squared minus 1 by, 1 + delta -1 ;; equals delta into 2 plus delta by delta ;; equals 2 + delta;; equals 2

Both the limits of f of x = x squared minus 1 by x minus 1 is shown. The right hand limit and left hand limit converge to 2.

Limit of a function at x = a is understood as the valie of function at x=a, left hand side of that : x = a minus delta, and right side of that x = a + delta.

Limits of a function at x = a are illustrated in the figure