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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
mathsLimit of a functionUnderstanding limits with Graphs

Understanding limits with the graph of the function

Geometrical meaning of finding limit of a function is explained.



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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
  • 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`.

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`

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)`

                            
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