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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

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mathsLimit of a functionUnderstanding limits with Graphs

### 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.

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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. The figure plots the numerator and denominator : 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.

The values of denominator at 1-delta and 1+delta are -delta and +delta respectively.

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.

The values of numerator at 1-delta and 1+delta are -2delta+delta^2 and +2delta+delta^2 respectively.

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.

It is noted that the slope of the line defines the values of denominator at 1-delta and 1+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.

It is noted that the slope of the line defines the values of numerator at 1-delta and 1+delta.

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)).

Slope of numerator and denominator defines the limits of the function.

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.

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