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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.
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.

click on the content to continue..

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. 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.

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.

•  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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