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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 functionLimit of Algebraic Expressions

### Limit of Polynomials

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

click on the content to continue..

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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