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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

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Limits involving Binomial Expressions

» Special case of canceling factors in numerator and denominator

→ `lim_(x->a)(x^n-a^n)/(x-a) = na^(n-1)`

» With change of variable `x=1+y` and constant `a=1`

→ `lim_(y->0) ((1+y)^n - 1)/y = n`

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

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

*simple steps to build the foundation*

trek

*simple steps to build the foundation*

trek

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

Starting on learning "Limits involving binomial expressions". ;; Factoring binomials is revised and the limit involving binomials is explained with an example.

Which of the following refers to factoring binomials?

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

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

What does binomial mean?

- 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`
- `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`
- `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)`

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**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)`

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

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

- `3/2`
- `2/3`
- `1`
- `0`

The answer is '`3/2`'

*your progress details*

Progress

*About you*

Progress

Which of the following refers to factoring binomials?

c;squared;sea;see

factoring a. + b + c squared

power;n;minus

factoring of a power n minus b power n

The answer is "factoring of a power n minus b power n"

What does binomial mean?

two;terms;2

of two terms

compound;bismuth

a compound of bismuth

The answer is "of two terms". ;; The expression a power n minus b power n is a two variate binomial of degree n. ;; two variate : a. and b are the two variables.;; binomial : a power n and b power n are the two terms ;; degree n : the maximum power is n.

factoring the binomials is given by : a power n minus b power n ;; equals a minus b, multiplied, a power n minus 1 + a power n minus 2 multiplied b + a power n minus 3 multiplied b power 2 + dot dot dot + b power n minus 1

What is the value of f of x = x power 4 minus 16, by x minus 2 at x =2?

by

0 by 0

1

1

0

0

infinity

infinity

The answer is "0 by 0"

What is the limit of f of x = x power 4 minus 16, by x minus 2 at x =2?

by

0 by 0

times;power;3;2;4

4 times 2 power 3

0

0

infinity

infinity

The answer is "4 times 2 power 3". The answer is explained in the next page.

Limit of f of x = x power 4 minus 16, by x minus 2 at x =2. On substitution of x=2, the function evaluates to 0 by 0;; limit of the function is ;; limit x tending to 2, x power 4 minus 16, by x minus 2 ;; equals limit x tending to 2, x power 4 minus 2 power 4, by x minus 2 ;; equals limit x tending to 2, x minus 2 multiplied x cube + 2 x squared + 4 x + 8, divided by, x minus 2;; equals limit x tending to 2, x cube + 2 x squared + 4 x + 8;; equals 2 power 3 + 2 times 2 power 2+ 4 times 2 + 8;; equals 4 into 2 power 3.;; Note : x minus 2 is not canceled at x=2. But, limit is applied to x minus 2 by x minus 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; limit x tending to a. x power n minus a power n divided by x minus a, = , n a power n minus 1.

What is the limit for f of x equal x power 15 minus 1, by x power 10 minus 1, at x=1?

1

3 by 2

2

2 by 3

3

4

The answer is "3 by 2"