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

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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`'

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