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Limit of Algebraic Expressions

Limit of Algebraic Expressions

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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`

Limits involving Binomial Expressions

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.


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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`'

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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"

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