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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 functionCalculating Limits

### Limit of continuous functions

With an example, calculation of limits for a continuous function is discussed. The condition under which a function is continuous is illustrated with examples.

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Given function f(x)=1/(x^2+1). what is f(x)|_(x=1)?

• Substitute x=1
• 1/(1+1)
• 1/2
• all the above
• all the above

The answer is 'All the above'. The options provide the steps to evaluate f(1).

Given function f(x)=1/(x^2+1). what is left-hand-limit lim_(x->1-)f(x)?

• Substitute x=1
• Substitute x=1+delta
• Substitute x=1-delta
• Substitute x=1-delta
• all the above

The answer is 'Substitute x=1-delta'.

lim_(x->1-)f(x)
quad quad = 1/((1-delta)^2+1)
quad quad = 1/(1-2delta+delta^2+1)
quad quad = 1/(2-2delta+delta^2)
quad quad = 1/2 quad quad quad quad (substituting delta=0)

Given function f(x)=1/(x^2+1). what is right-hand-limit lim_(x->1+)f(x)?

• Substitute x=1
• Substitute x=1+delta
• Substitute x=1+delta
• Substitute x=1-delta
• all the above

The answer is 'Substitute x=1+delta'.

lim_(x->1+)f(x)
quad quad = 1/((1+delta)^2+1)
quad quad = 1/(1+2delta+delta^2+1)
quad quad = 1/(2+2delta+delta^2)
quad quad = 1/2 quad quad quad quad (substituting delta=0)

Given function f(x)=1/(x^2+1).
•  f(1)=1/2
•  lim_(x->1-)f(x) = 1/2
•  lim_(x->1+)f(x) = 1/2
The function is continuous at x=1.

Note: Though, as part of the topic, the functions that are discontinuous or indeterminate or non-differentiable are mainly handled, finding limits is not only for those kind of functions. Any function at any point can be evaluated for limits.

Function is continuous if all three values are equal.

Function is continuous: if f(a) = lim_(x->a-)f(x) =lim_(x->a+)f(x), then the function is continuous at x=a.

Solved Exercise Problem:

Given function f(x) = sin x, is it continuous at input values x= 0, pi/2, pi?

• Continuous at 0 and pi/2 only
• Continuous at all three input values
• Continuous at all three input values
• Continuous at only 0
• not continuous at any one of the given values

The answer is 'Continuous at all three input values'

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