nubtrek

Server Error

Server Not Reachable.

This may be due to your internet connection or the nubtrek server is offline.

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
mathsLimit of a functionLimit of Algebraic Expressions

Limits involving Binomial Expressions

Factoring binomials is revised and the limit involving binomials is explained with an example.



click on the content to continue..

Which of the following refers to factoring binomials?

  • factoring `(a+b+c)^2`
  • factoring of `a^n-b^n`
  • factoring of `a^n-b^n`

The answer is 'factoring of `a^n-b^n`'

What does binomial mean?

  • of two terms
  • of two terms
  • a compound of bismuth

The answer is 'of two terms'.

The expression `a^n - b^n` is a two variate binomial of degree `n`.
two variate : `a` and `b` are two variables
binomial : `a^n` and `-b^n` are the two terms
degree `n`: The maximum power is `n`

Factoring the binomials is given by
`a^n-b^n`
`quad quad = (a-b)(a^(n-1)+a^(n-2)b^1+`
`quad quad quad quad a^(n-3)b^2+ cdots + b^(n-1))`

What is the value of `f(x)=color(deepskyblue)(x^4-16)/color(coral)(x-2)` at `x=2`?

  • `0/0`
  • `0/0`
  • `1`
  • `0`
  • `oo`

The answer is '`0/0`'

What is the limit of `f(x)=color(deepskyblue)(x^4-16)/color(coral)(x-2)` at `x=2`?

  • `0/0`
  • `4 xx 2^3`
  • `4 xx 2^3`
  • `0`
  • `oo`

The answer is '`4 xx 2^3`'. The answer is explained in the next page.

limit of `f(x)=color(deepskyblue)(x^4-16)/color(coral)(x-2)` at `x=2`.
On substitution of `x=2` the function evaluates to `0/0`

Limit of the function is
`lim_(x->2) color(deepskyblue)(x^4-16)/color(coral)(x-2)`
`quad quad = lim_(x->2) color(deepskyblue)(x^4-2^4)/color(coral)(x-2)`
`quad quad = lim_(x->2) color(deepskyblue)((x-2)(x^3+2x^2+4x+8))/color(coral)(x-2)`
`quad quad = lim_(x->2) (x^3+2x^2+4x+8)`
`quad quad = 2^3+ 2xx 2^2 + 4xx 2 + 8`
`quad quad = 4 xx 2^3`

note: `(x-2)` is not canceled at `x=2`. But, limit is applied to `(x-2)/(x-2)`

To find limit of functions with binomials, factor the binomials to cancel out the factor involving `0`.

Limit of expressions with Binomials: For any positive integer n
`lim_(x->a)color(deepskyblue)(x^n-a^n)/color(coral)(x-a) = na^(n-1)`

Solved Exercise Problem:

What is the limit for `f(x)=color(deepskyblue)(x^15-1)/color(coral)(x^10-1)` at `x=1`

  • `3/2`
  • `3/2`
  • `2/3`
  • `1`
  • `0`

The answer is '`3/2`'

                            
switch to slide-show version