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Thought-Process to Discover Knowledge

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  nub,

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nub is the simple explanation of the concept.

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exercise provides practice problems to become fluent in the concepts.

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summary of this topic

Understanding limits with Graphs

Understanding limits with Graphs

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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)`

Understanding limits with the graph of the function

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

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Geometrical meaning of finding limit of a function is explained.


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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.limit of a defined function 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.left-hand-limit of a defined function 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.right-hand-limit of a defined function 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.limits of a defined function 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.

limits of a defined function  •  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

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