Calculus - Limit : Limits involving Binomial Expressions

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Limits involving Binomial Expressions

what you'll learn...

Limits involving Binomial Expressions

» Special case of canceling factors in numerator and denominator
→ $lim x \to a \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}$ $lim_(x->a)(x^n-a^n)/(x-a) = na^(n-1)$

» With change of variable $x = 1 + y$ $x=1+y$ and constant $a = 1$ $a=1$
→ $lim y \to 0 \frac{{(1 + y)}^{n} - 1}{y} = n$ $lim_(y->0) ((1+y)^n - 1)/y = n$

recap

Factoring binomials refers to factoring of $a^{n} - b^{n}$ $a^n-b^n$ .

The word binomial means: of two terms.

The expression $a^{n} - b^{n}$ $a^n - b^n$ 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
$a^n-b^n$
$quad quad = (a-b)(a^(n-1)+a^(n-2)b^1+$
$quad quad quad quad a^(n-3)b^2+ cdots + b^(n-1))$

example

The value of $f(x)=color(deepskyblue)(x^4-16)/color(coral)(x-2)$ at $x=2$ is $0/0$ by direct substitution $x=2$ .

The limit of $f(x)=color(deepskyblue)(x^4-16)/color(coral)(x-2)$ at $x=2$ is $4 xx 2^3$ .

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 $0$ .

What is the limit for $f(x)=color(deepskyblue)(x^15-1)/color(coral)(x^10-1)$ at $x=1$
The answer is ' $3/2$ '

summary

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

Outline

The outline of material to learn "limits (calculus)" is as follows.

Note : click here for detailed outline of Limits(Calculus).

→ Indeterminate and Undefined

→ Indeterminate value in Functions

→ Expected Value

→ Continuity

→ Definition by Limits

→ Geometrical Explanation for Limits

→ Limit with Numerator and Denominator

→ Limits of Ratios - Examples

→ L'hospital Rule

→ Examining a function

→ Algebra of Limits

→ Limit of a Polynomial

→ Limit of Ratio of Zeros

→ Limit of ratio of infinities

→ limit of Binomial

→ Limit of Non-algebraic Functions