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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

Finding limit of standard polynomial expressions is explained with an example.



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How to find limits of function `f(x)=3x^3+2x^2-1` at `x=-2`

  • apply limit to the three terms of the function
  • apply limit to the three terms of the function
  • one has to memorize a formula

The answer is 'apply limit to the three terms of the function'.

This is explained in the next page.

limit of function `f(x)=color(deepskyblue)(3x^3)+color(coral)(2x^2)-1` at `x=-2`

By Substitution :
`f(x)|_(x=-2)`
`quad quad = color(deepskyblue)(3(-2)^3)+color(coral)(2(-2)^2)-1`
`quad quad = color(deepskyblue)(3(-8))+color(coral)(2(4))-1`
`quad quad = -17`

limits of function `f(x)=color(deepskyblue)(3x^3)+color(coral)(2x^2)-1` at `x=-2`

left-hand-limit :
`lim_(x->-2-)f(x)`
`quad quad = color(deepskyblue)(3(-2-delta)^3)+color(coral)(2(-2-delta)^2)-1`
substitute `delta=0`
`quad quad = color(deepskyblue)(3(-8))+color(coral)(2(4))-1`
`quad quad = -17`

limits of function `f(x)=color(deepskyblue)(3x^3)+color(coral)(2x^2)-1` at `x=-2`

right-hand-limit :
`lim_(x->-2+)f(x)`
`quad quad = color(deepskyblue)(3(-2+delta)^3)+color(coral)(2(-2+delta)^2)-1`
substitute `delta=0`
`quad quad = color(deepskyblue)(3(-8))+color(coral)(2(4))-1`
`quad quad = -17`

limits of function `f(x)=3x^3+2x^2-1` at `x=-2`

`f(x)|_(x=-2) = -17`

`lim_(x->-2-)f(x)= -17`

`lim_(x->-2+)f(x)= -17`

All three values are equal. So the function is continuous.

Limit of a polynomial is the limit on individual terms of the polynomial.

Limit of a polynomial: For a function `f(x) = a_nx^n+a_(n-1)x^(n-1)+ cdots + a_1x^1 + a_0`
`lim_(x->a) f(x) `
`quad quad = a_n lim_(x->a) x^n+a_(n-1) lim_(x->a) x^(n-1)+ cdots `
`quad quad quad quad + a_1 lim_(x->a) x^1 + a_0`

                            
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